Téglalapszámok – Wikipédia

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data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Tartalomjegyzék</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">elrejtés</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">Bevezető</div> </a> </li> <li id="toc-Figurális_számokként" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Figurális_számokként"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Figurális számokként</span> </div> </a> <ul id="toc-Figurális_számokként-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Első_n_téglalapszám_összege" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Első_n_téglalapszám_összege"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Első <span>n</span> téglalapszám összege</span> </div> </a> <ul id="toc-Első_n_téglalapszám_összege-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reciprokösszegek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Reciprokösszegek"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Reciprokösszegek</span> </div> </a> <button aria-controls="toc-Reciprokösszegek-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>A(z) Reciprokösszegek alszakasz kinyitása/becsukása</span> </button> <ul id="toc-Reciprokösszegek-sublist" class="vector-toc-list"> <li id="toc-Általánosítás" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Általánosítás"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Általánosítás</span> </div> </a> <ul id="toc-Általánosítás-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-További_tulajdonságaik" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#További_tulajdonságaik"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>További tulajdonságaik</span> </div> </a> <ul id="toc-További_tulajdonságaik-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-További_információk" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#További_információk"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>További információk</span> </div> </a> <ul id="toc-További_információk-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fordítás" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Fordítás"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Fordítás</span> </div> </a> <ul id="toc-Fordítás-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jegyzetek" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Jegyzetek"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Jegyzetek</span> </div> </a> <ul id="toc-Jegyzetek-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Tartalomjegyzék" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Tartalomjegyzék kinyitása/becsukása" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Tartalomjegyzék kinyitása/becsukása</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Téglalapszámok</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Ugrás egy más nyelvű szócikkre. Elérhető 19 nyelven" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 nyelv</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Pronic_number" title="Pronic number – angol" lang="en" hreflang="en" data-title="Pronic number" data-language-autonym="English" data-language-local-name="angol" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%B9%D8%AF%D8%AF_%D8%A8%D8%B1%D9%88%D9%86%D9%8A" title="عدد بروني – arab" lang="ar" hreflang="ar" data-title="عدد بروني" data-language-autonym="العربية" data-language-local-name="arab" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D0%B0%D0%B2%D0%BE%D1%8A%D0%B3%D1%8A%D0%BB%D0%BD%D0%BE_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Правоъгълно число – bolgár" lang="bg" hreflang="bg" data-title="Правоъгълно число" data-language-autonym="Български" data-language-local-name="bolgár" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Nombre_rectangular" title="Nombre rectangular – katalán" lang="ca" hreflang="ca" data-title="Nombre rectangular" data-language-autonym="Català" data-language-local-name="katalán" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Rechteckzahl" title="Rechteckzahl – német" lang="de" hreflang="de" data-title="Rechteckzahl" data-language-autonym="Deutsch" data-language-local-name="német" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-eml mw-list-item"><a href="https://eml.wikipedia.org/wiki/N%C3%B9mer_obl%C3%B9ng" title="Nùmer oblùng – Emiliano-Romagnolo" lang="egl" hreflang="egl" data-title="Nùmer oblùng" data-language-autonym="Emiliàn e rumagnòl" data-language-local-name="Emiliano-Romagnolo" class="interlanguage-link-target"><span>Emiliàn e rumagnòl</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/N%C3%BAmero_oblongo" title="Número oblongo – spanyol" lang="es" hreflang="es" data-title="Número oblongo" data-language-autonym="Español" data-language-local-name="spanyol" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Nombre_oblong" title="Nombre oblong – francia" lang="fr" hreflang="fr" data-title="Nombre oblong" data-language-autonym="Français" data-language-local-name="francia" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Numero_oblungo" title="Numero oblungo – olasz" lang="it" hreflang="it" data-title="Numero oblungo" data-language-autonym="Italiano" data-language-local-name="olasz" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%9F%A9%E5%BD%A2%E6%95%B0" title="矩形数 – japán" lang="ja" hreflang="ja" data-title="矩形数" data-language-autonym="日本語" data-language-local-name="japán" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/N%C3%BAmero_oblongo" title="Número oblongo – portugál" lang="pt" hreflang="pt" data-title="Número oblongo" data-language-autonym="Português" data-language-local-name="portugál" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Num%C4%83r_rectangular" title="Număr rectangular – román" lang="ro" hreflang="ro" data-title="Număr rectangular" data-language-autonym="Română" data-language-local-name="román" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D1%80%D1%8F%D0%BC%D0%BE%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%BE%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Прямоугольное число – orosz" lang="ru" hreflang="ru" data-title="Прямоугольное число" data-language-autonym="Русский" data-language-local-name="orosz" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Podol%C5%BEno_%C5%A1tevilo" title="Podolžno število – szlovén" lang="sl" hreflang="sl" data-title="Podolžno število" data-language-autonym="Slovenščina" data-language-local-name="szlovén" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Rektangeltal" title="Rektangeltal – svéd" lang="sv" hreflang="sv" data-title="Rektangeltal" data-language-autonym="Svenska" data-language-local-name="svéd" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%9A%E0%AF%86%E0%AE%B5%E0%AF%8D%E0%AE%B5%E0%AE%95_%E0%AE%8E%E0%AE%A3%E0%AF%8D" title="செவ்வக எண் – tamil" lang="ta" hreflang="ta" data-title="செவ்வக எண்" data-language-autonym="தமிழ்" data-language-local-name="tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D1%80%D1%8F%D0%BC%D0%BE%D0%BA%D1%83%D1%82%D0%BD%D0%B5_%D1%87%D0%B8%D1%81%D0%BB%D0%BE" title="Прямокутне число – ukrán" lang="uk" hreflang="uk" data-title="Прямокутне число" data-language-autonym="Українська" data-language-local-name="ukrán" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B0" title="普洛尼克数 – wu kínai" lang="wuu" hreflang="wuu" data-title="普洛尼克数" data-language-autonym="吴语" data-language-local-name="wu kínai" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%99%AE%E6%B4%9B%E5%B0%BC%E5%85%8B%E6%95%B0" title="普洛尼克数 – kínai" lang="zh" hreflang="zh" data-title="普洛尼克数" data-language-autonym="中文" data-language-local-name="kínai" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1486643#sitelinks-wikipedia" title="Nyelvközi hivatkozások szerkesztése" class="wbc-editpage">Hivatkozások szerkesztése</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Névterek"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/T%C3%A9glalapsz%C3%A1mok" title="A lap megtekintése [c]" accesskey="c"><span>Szócikk</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Vita:T%C3%A9glalapsz%C3%A1mok" rel="discussion" title="Az oldal tartalmának megvitatása [t]" accesskey="t"><span>Vitalap</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Nyelvvariáns váltása" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">magyar</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Nézetek"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/T%C3%A9glalapsz%C3%A1mok"><span>Olvasás</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=history" title="A lap korábbi változatai [h]" accesskey="h"><span>Laptörténet</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Oldal eszközök"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Eszközök" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Eszközök</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Eszközök</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">áthelyezés az oldalsávba</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">elrejtés</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="További lehetőségek" > <div class="vector-menu-heading"> Műveletek </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/T%C3%A9glalapsz%C3%A1mok"><span>Olvasás</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit" title="Az oldal forráskódjának szerkesztése [e]" accesskey="e"><span>Szerkesztés</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=history"><span>Laptörténet</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> Általános </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Speci%C3%A1lis:Mi_hivatkozik_erre/T%C3%A9glalapsz%C3%A1mok" title="Az erre a lapra hivatkozó más lapok listája [j]" 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href="/w/index.php?title=Speci%C3%A1lis:Hivatkoz%C3%A1s&page=T%C3%A9glalapsz%C3%A1mok&id=25352880&wpFormIdentifier=titleform" title="Információk a lap idézésével kapcsolatban"><span>Hogyan hivatkozz erre a lapra?</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:UrlShortener&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FT%25C3%25A9glalapsz%25C3%25A1mok"><span>Rövidített URL készítése</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Speci%C3%A1lis:QrCode&url=https%3A%2F%2Fhu.wikipedia.org%2Fwiki%2FT%25C3%25A9glalapsz%25C3%25A1mok"><span>QR-kód letöltése</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Nyomtatás/exportálás </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-create_a_book" class="mw-list-item"><a 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id="mw-fr-revision-details" class="mw-fr-revision-details-dialog" style="display:none;"><div tabindex="0"></div><div class="cdx-dialog cdx-dialog--horizontal-actions"><header class="cdx-dialog__header cdx-dialog__header--default"><div class="cdx-dialog__header__title-group"><h2 class="cdx-dialog__header__title">Változat állapota</h2><p class="cdx-dialog__header__subtitle">Ez a lap egy ellenőrzött változata</p></div><button class="cdx-button cdx-button--action-default cdx-button--weight-quiet
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A téglalapszámok általánosíthatók az <span class="texhtml"><i>n</i>(<i>n</i> + k)</span> alakú számokra. </p><p>Az első néhány téglalapszám: </p> <dl><dd><a href="/wiki/0_(sz%C3%A1m)" title="0 (szám)">0</a>, <a href="/wiki/2_(sz%C3%A1m)" title="2 (szám)">2</a>, <a href="/wiki/6_(sz%C3%A1m)" title="6 (szám)">6</a>, <a href="/wiki/12_(sz%C3%A1m)" title="12 (szám)">12</a>, <a href="/wiki/20_(sz%C3%A1m)" title="20 (szám)">20</a>, <a href="/wiki/30_(sz%C3%A1m)" title="30 (szám)">30</a>, <a href="/wiki/42_(sz%C3%A1m)" title="42 (szám)">42</a>, <a href="/wiki/56_(sz%C3%A1m)" title="56 (szám)">56</a>, <a href="/wiki/72_(sz%C3%A1m)" title="72 (szám)">72</a>, <a href="/wiki/90_(sz%C3%A1m)" title="90 (szám)">90</a>, <a href="/wiki/110_(sz%C3%A1m)" title="110 (szám)">110</a>, <a href="/wiki/132_(sz%C3%A1m)" title="132 (szám)">132</a>, <a href="/wiki/156_(sz%C3%A1m)" title="156 (szám)">156</a>, <a href="/wiki/182_(sz%C3%A1m)" title="182 (szám)">182</a>, <a href="/wiki/210_(sz%C3%A1m)" title="210 (szám)">210</a>, <a href="/wiki/240_(sz%C3%A1m)" title="240 (szám)">240</a>, <a href="/wiki/272_(sz%C3%A1m)" title="272 (szám)">272</a>, <a href="/wiki/206_(sz%C3%A1m)" title="206 (szám)">306</a>, <a href="/wiki/342_(sz%C3%A1m)" title="342 (szám)">342</a>, <a href="/wiki/380_(sz%C3%A1m)" title="380 (szám)">380</a>, <a href="/wiki/420_(sz%C3%A1m)" title="420 (szám)">420</a>, <a href="/wiki/462_(sz%C3%A1m)" title="462 (szám)">462</a> … (<a rel="nofollow" class="external text" href="//oeis.org/A002378">A002378</a> sorozat az <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>-ben).</dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Figurális_számokként"><span id="Figur.C3.A1lis_sz.C3.A1mokk.C3.A9nt"></span>Figurális számokként</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=1" title="Szakasz szerkesztése: Figurális számokként"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Arisztotel%C3%A9sz_metafizik%C3%A1ja" class="mw-redirect" title="Arisztotelész metafizikája">Arisztotelész metafizikájában</a> a téglalapszámokat más <a href="/wiki/Figur%C3%A1lis_sz%C3%A1mok" title="Figurális számok">figurális számokkal</a>, a <a href="/wiki/H%C3%A1romsz%C3%B6gsz%C3%A1mok" title="Háromszögszámok">háromszögszámokkal</a> és <a href="/wiki/N%C3%A9gyzetsz%C3%A1mok" title="Négyzetszámok">négyzetszámokkal</a> együtt tanulmányozták,<sup id="cite_ref-knorr_2-0" class="reference"><a href="#cite_note-knorr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> felfedezésük még korábbra, a <a href="/wiki/P%C3%BCthagoreusok" title="Püthagoreusok">püthagoreusokhoz</a> köthető.<sup id="cite_ref-hist_3-0" class="reference"><a href="#cite_note-hist-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> A sokszögszámok mintájára: </p> <dl><dd><table style="text-align: center"> <tbody><tr valign="bottom"> <td style="padding: 0 1em"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td style="padding: 0 1em"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td style="padding: 0 1em"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td> <td style="padding: 0 1em"><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span><br /><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <span typeof="mw:File"><a href="/wiki/F%C3%A1jl:GrayDot.svg" class="mw-file-description" title="*"><img alt="*" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/16px-GrayDot.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/24px-GrayDot.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/GrayDot.svg/32px-GrayDot.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> </td></tr> <tr> <td>1×2</td> <td>2×3</td> <td>3×4</td> <td>4×5 </td></tr></tbody></table></dd></dl> <p>Az <span class="texhtml mvar" style="font-style:italic;">n</span>-edik téglalapszám épp kétszerese az <span class="texhtml mvar" style="font-style:italic;">n</span>-edik <a href="/wiki/H%C3%A1romsz%C3%B6gsz%C3%A1m" class="mw-redirect" title="Háromszögszám">háromszögszámnak</a><sup id="cite_ref-bon_1-1" class="reference"><a href="#cite_note-bon-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-knorr_2-1" class="reference"><a href="#cite_note-knorr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> és <span class="texhtml mvar" style="font-style:italic;">n</span>-nel haladja meg az <span class="texhtml mvar" style="font-style:italic;">n</span>-edik <a href="/wiki/N%C3%A9gyzetsz%C3%A1m" class="mw-redirect" title="Négyzetszám">négyzetszámot</a>, ami az alternatív <span class="texhtml"><i>n</i><sup>2</sup> + <i>n</i></span> képletükből is világos. Az <span class="texhtml mvar" style="font-style:italic;">n</span>-edik téglalapszám éppen a páratlan négyzetszám <span class="texhtml">(2<i>n</i> + 1)<sup>2</sup></span> és az <span class="texhtml">(<i>n</i>+1)</span>-edik <a href="/wiki/K%C3%B6z%C3%A9ppontos_hatsz%C3%B6gsz%C3%A1m" class="mw-redirect" title="Középpontos hatszögszám">középpontos hatszögszám</a> közötti különbség. </p> <div class="mw-heading mw-heading2"><h2 id="Első_n_téglalapszám_összege"><span id="Els.C5.91_n_t.C3.A9glalapsz.C3.A1m_.C3.B6sszege"></span>Első <span class="texhtml mvar" style="font-style:italic;">n</span> téglalapszám összege</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=2" title="Szakasz szerkesztése: Első n téglalapszám összege"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/F%C3%A1jl:Pronic_partial_sum.png" class="mw-file-description"><img alt="A téglalapszámok egy részösszegének vizuális ábrázolása" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Pronic_partial_sum.png/440px-Pronic_partial_sum.png" decoding="async" width="440" height="139" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Pronic_partial_sum.png/660px-Pronic_partial_sum.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/fc/Pronic_partial_sum.png 2x" data-file-width="791" data-file-height="250" /></a><figcaption>A téglalapszámok egy részösszegének vizuális ábrázolása</figcaption></figure> <p>A téglalapszámok figurális mivoltuk miatt a legegyszerűbben téglalapokként ábrázolhatóak, ahogyan az ábrán látható. Az első <span class="texhtml mvar" style="font-style:italic;">n</span> téglalapszám összegét meghatározhatjuk, ha a nagy téglalap területéből kivonjuk a nem kellő területeket. </p><p>A nagy téglalap területe <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+2+\cdots +n)\cdot (n+1)={\frac {n(n+1)^{2}}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+2+\cdots +n)\cdot (n+1)={\frac {n(n+1)^{2}}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ec286f7567778f103921743b2faeb7be30fd84b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:39.25ex; height:5.843ex;" alt="{\displaystyle (1+2+\cdots +n)\cdot (n+1)={\frac {n(n+1)^{2}}{2}}}"></span>. </p><p>Megfigyelhető, hogy a felesleges részek területei soronként az első 1, 2, ..., n-1 pozitív szám összegei, azaz <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)+(1+2)+\cdots +(1+2+\cdots +n-1)={\frac {1(2)}{2}}+{\frac {2(3)}{2}}+\cdots +{\frac {n(n-1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>1</mn> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)+(1+2)+\cdots +(1+2+\cdots +n-1)={\frac {1(2)}{2}}+{\frac {2(3)}{2}}+\cdots +{\frac {n(n-1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5e4c8c30ac5e33b423c6e391db62d71c542e36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:75.688ex; height:5.676ex;" alt="{\displaystyle (1)+(1+2)+\cdots +(1+2+\cdots +n-1)={\frac {1(2)}{2}}+{\frac {2(3)}{2}}+\cdots +{\frac {n(n-1)}{2}}}"></span>. </p><p>Továbbá látható, hogy a felesleges részek pontosan az első <span class="texhtml mvar" style="font-style:italic;">n-1</span> téglalapszám összegének a fele. </p><p>Ekkor ha <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1c49fad1eccc4e9af1e4f23f32efdc3ac4da973" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.483ex; height:2.843ex;" alt="{\displaystyle f(n)}"></span> az első <span class="texhtml mvar" style="font-style:italic;">n</span> téglalapszám összegét adja meg, akkor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\frac {n(n+1)^{2}}{2}}-{\frac {f(n-1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\frac {n(n+1)^{2}}{2}}-{\frac {f(n-1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/353489237eeef26e4996ff22cc01f3d578960b03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:30.235ex; height:5.843ex;" alt="{\displaystyle f(n)={\frac {n(n+1)^{2}}{2}}-{\frac {f(n-1)}{2}}}"></span>. </p><p>Felhasználva, hogy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)=f(n-1)+n(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)=f(n-1)+n(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0e99ad0b3de212271dd5fc6e839be794c6b95ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.508ex; height:2.843ex;" alt="{\displaystyle f(n)=f(n-1)+n(n+1)}"></span> és az algebra szabályait segítségül hívva: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\frac {n(n+1)^{2}-f(n)+n(n+1)}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\frac {n(n+1)^{2}-f(n)+n(n+1)}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29d8ff4b89ce95e79f00e5ffcd518b89a6f843a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.838ex; height:5.843ex;" alt="{\displaystyle f(n)={\frac {n(n+1)^{2}-f(n)+n(n+1)}{2}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2f(n)=n(n+1)^{2}-f(n)+n(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2f(n)=n(n+1)^{2}-f(n)+n(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7178d188299e97d903ae4d6e924afaa044b461de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.164ex; height:3.176ex;" alt="{\displaystyle 2f(n)=n(n+1)^{2}-f(n)+n(n+1)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3f(n)=n(n+1)^{2}+n(n+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3f(n)=n(n+1)^{2}+n(n+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d36af9f427d72d6288beb8e7c19115409a7cd759" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.841ex; height:3.176ex;" alt="{\displaystyle 3f(n)=n(n+1)^{2}+n(n+1)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3f(n)=n(n+1)(n+1+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3f(n)=n(n+1)(n+1+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/788073cad8a9c3deb7f0d1d1627437c4e7d2ef11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.555ex; height:2.843ex;" alt="{\displaystyle 3f(n)=n(n+1)(n+1+1)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(n)={\frac {n(n+1)(n+2)}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(n)={\frac {n(n+1)(n+2)}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971e689d2eb06bfda9918b299040c955476e1b8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.226ex; height:5.676ex;" alt="{\displaystyle f(n)={\frac {n(n+1)(n+2)}{3}}}"></span></dd></dl> <p>Azaz </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}i(i+1)={\frac {n(n+1)(n+2)}{3}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mn>3</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}i(i+1)={\frac {n(n+1)(n+2)}{3}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b73f32bf4563e4952edc6eb43c6d11c42aeb69e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.902ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}i(i+1)={\frac {n(n+1)(n+2)}{3}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Reciprokösszegek"><span id="Reciprok.C3.B6sszegek"></span>Reciprokösszegek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=3" title="Szakasz szerkesztése: Reciprokösszegek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az első <span class="texhtml mvar" style="font-style:italic;">n</span> pozitív téglalapszám reciprokösszege a következőképpen alakul: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{(n-1)\cdot n}}+{\frac {1}{n\cdot (n+1)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>⋅</mo> <mn>3</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <mn>4</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{(n-1)\cdot n}}+{\frac {1}{n\cdot (n+1)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1100d79a2c6f7200a8048b00c8d6393bff8534e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:53.679ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{1\cdot 2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{3\cdot 4}}+\cdots +{\frac {1}{(n-1)\cdot n}}+{\frac {1}{n\cdot (n+1)}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =\left({\frac {1}{1}}-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{n-1}}-{\frac {1}{n}}\right)+\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =\left({\frac {1}{1}}-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{n-1}}-{\frac {1}{n}}\right)+\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf52e3da1a665321c0c77f300afaf8cb62542a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:66.508ex; height:6.176ex;" alt="{\displaystyle =\left({\frac {1}{1}}-{\frac {1}{2}}\right)+\left({\frac {1}{2}}-{\frac {1}{3}}\right)+\cdots +\left({\frac {1}{n-1}}-{\frac {1}{n}}\right)+\left({\frac {1}{n}}-{\frac {1}{n+1}}\right)}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{1}}+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{n-1}}+{\frac {1}{n-1}}\right)+\left(-{\frac {1}{n}}+{\frac {1}{n}}\right)-{\frac {1}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{1}}+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{n-1}}+{\frac {1}{n-1}}\right)+\left(-{\frac {1}{n}}+{\frac {1}{n}}\right)-{\frac {1}{n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/265ba9c249bbad698def223f1c658115b411807c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:87.653ex; height:6.176ex;" alt="{\displaystyle ={\frac {1}{1}}+\left(-{\frac {1}{2}}+{\frac {1}{2}}\right)+\left(-{\frac {1}{3}}+{\frac {1}{3}}\right)+\cdots +\left(-{\frac {1}{n-1}}+{\frac {1}{n-1}}\right)+\left(-{\frac {1}{n}}+{\frac {1}{n}}\right)-{\frac {1}{n+1}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle =1-{\frac {1}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =1-{\frac {1}{n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d18f6b10a848a3b11baefcef9818491e8c10bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:12.69ex; height:5.343ex;" alt="{\displaystyle =1-{\frac {1}{n+1}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {n}{n+1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {n}{n+1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bde641e0cbc372225f0d92aa259e448eb6e4e406" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.687ex; height:4.843ex;" alt="{\displaystyle ={\frac {n}{n+1}}}"></span></dd></dl> <p>Ebből kifolyólag a pozitív téglalapszámok reciprokösszege 1:<sup id="cite_ref-telescope_4-0" class="reference"><a href="#cite_note-telescope-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}=\lim _{n\rightarrow \infty }{\frac {n}{n+1}}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}=\lim _{n\rightarrow \infty }{\frac {n}{n+1}}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a556912b224463e48fb72cde7737b3f9fa84fac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.248ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+1)}}=\lim _{n\rightarrow \infty }{\frac {n}{n+1}}=1}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Általánosítás"><span id=".C3.81ltal.C3.A1nos.C3.ADt.C3.A1s"></span>Általánosítás</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=4" title="Szakasz szerkesztése: Általánosítás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A téglalapszámok általánosíthatóak <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(n+k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(n+k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c6de32979e0c07b117c56c4dc307ca8f1ec30d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.65ex; height:2.843ex;" alt="{\displaystyle n(n+k)}"></span> alakúra, ahol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {Z} ^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {Z} ^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b871e54d85bc00d974ec76edc49f3b71a489cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.113ex; height:2.509ex;" alt="{\displaystyle k\in \mathbb {Z} ^{+}}"></span>. Ebben az esetben az első <span class="texhtml mvar" style="font-style:italic;">n</span> pozitív téglalapszám reciprokösszege a következő: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{1\cdot (1+k)}}+{\frac {1}{2\cdot (2+k)}}+{\frac {1}{3\cdot (3+k)}}+\cdots +{\frac {1}{(n-1)\cdot (n+k-1)}}+{\frac {1}{n\cdot (n+k)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mo>⋅</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{1\cdot (1+k)}}+{\frac {1}{2\cdot (2+k)}}+{\frac {1}{3\cdot (3+k)}}+\cdots +{\frac {1}{(n-1)\cdot (n+k-1)}}+{\frac {1}{n\cdot (n+k)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05a2eec25bbb2e8aa213169208bf9435dd3cc521" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:81.175ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{1\cdot (1+k)}}+{\frac {1}{2\cdot (2+k)}}+{\frac {1}{3\cdot (3+k)}}+\cdots +{\frac {1}{(n-1)\cdot (n+k-1)}}+{\frac {1}{n\cdot (n+k)}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}\left({\frac {1}{1}}-{\frac {1}{1+k}}\right)+\left({\frac {1}{2}}-{\frac {1}{2+k}}\right)+\cdots +\left({\frac {1}{n-1}}-{\frac {1}{n+k-1}}\right)+\left({\frac {1}{n}}-{\frac {1}{n+k}}\right){\Biggr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">[</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}\left({\frac {1}{1}}-{\frac {1}{1+k}}\right)+\left({\frac {1}{2}}-{\frac {1}{2+k}}\right)+\cdots +\left({\frac {1}{n-1}}-{\frac {1}{n+k-1}}\right)+\left({\frac {1}{n}}-{\frac {1}{n+k}}\right){\Biggr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90c9b3a99ad5608b9724b0a4c20faf7b1df16408" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:89.925ex; height:7.509ex;" alt="{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}\left({\frac {1}{1}}-{\frac {1}{1+k}}\right)+\left({\frac {1}{2}}-{\frac {1}{2+k}}\right)+\cdots +\left({\frac {1}{n-1}}-{\frac {1}{n+k-1}}\right)+\left({\frac {1}{n}}-{\frac {1}{n+k}}\right){\Biggr ]}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}\left({\frac {1}{1}}+{\frac {1}{2}}+\cdots +{\frac {1}{n}}\right)+\left(-{\frac {1}{1+k}}-{\frac {1}{2+k}}-\cdots -{\frac {1}{n+k}}\right){\Biggr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">[</mo> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>1</mn> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}\left({\frac {1}{1}}+{\frac {1}{2}}+\cdots +{\frac {1}{n}}\right)+\left(-{\frac {1}{1+k}}-{\frac {1}{2+k}}-\cdots -{\frac {1}{n+k}}\right){\Biggr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21d3d6c652b64e4bfbecff0dc15b388e56d84254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:68.255ex; height:7.509ex;" alt="{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}\left({\frac {1}{1}}+{\frac {1}{2}}+\cdots +{\frac {1}{n}}\right)+\left(-{\frac {1}{1+k}}-{\frac {1}{2+k}}-\cdots -{\frac {1}{n+k}}\right){\Biggr ]}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}H_{n}-\left(H_{n+k}-H_{k}\right){\Biggr ]}={\frac {1}{k}}\cdot {\Biggr [}H_{k}+H_{n}-H_{n+k}{\Biggr ]}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">[</mo> </mrow> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">[</mo> </mrow> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}H_{n}-\left(H_{n+k}-H_{k}\right){\Biggr ]}={\frac {1}{k}}\cdot {\Biggr [}H_{k}+H_{n}-H_{n+k}{\Biggr ]}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e57a0f0376c893bcc1e1dcb8fd63499030c4cb02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:54.892ex; height:7.509ex;" alt="{\displaystyle ={\frac {1}{k}}\cdot {\Biggr [}H_{n}-\left(H_{n+k}-H_{k}\right){\Biggr ]}={\frac {1}{k}}\cdot {\Biggr [}H_{k}+H_{n}-H_{n+k}{\Biggr ]}}"></span></dd></dl> <p>ahol a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e63458b04288bbe116a9a8037dfae0b36b2c639a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.15ex; height:2.509ex;" alt="{\displaystyle H_{n}}"></span> az első <span class="texhtml mvar" style="font-style:italic;">n</span> pozitív egész szám reciprokainak összegét, azaz az <span class="texhtml mvar" style="font-style:italic;">n</span>-dik <a href="/wiki/Harmonikus_sz%C3%A1m" title="Harmonikus szám">harmonikus számot</a> adja meg. </p><p>Ezen összeg <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9702f04f2d0e5b887b99faeeffb0c4cfd8263eee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.333ex; height:1.843ex;" alt="{\displaystyle n\rightarrow \infty }"></span> esetben: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+k)}}=\lim _{n\rightarrow \infty }{\frac {1}{k}}\cdot {\Biggr [}H_{k}+H_{n}-H_{n+k}{\Biggr ]}={\frac {1}{k}}\cdot {\Biggr [}\lim _{n\rightarrow \infty }H_{k}+\lim _{n\rightarrow \infty }\left(H_{n}-H_{n+k}\right){\Biggr ]}={\frac {H_{k}}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>i</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">[</mo> </mrow> </mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">[</mo> </mrow> </mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mi>k</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.470em" minsize="2.470em">]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+k)}}=\lim _{n\rightarrow \infty }{\frac {1}{k}}\cdot {\Biggr [}H_{k}+H_{n}-H_{n+k}{\Biggr ]}={\frac {1}{k}}\cdot {\Biggr [}\lim _{n\rightarrow \infty }H_{k}+\lim _{n\rightarrow \infty }\left(H_{n}-H_{n+k}\right){\Biggr ]}={\frac {H_{k}}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c74b40e29859c44bc34210f0ed52eed528574b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:88.902ex; height:7.509ex;" alt="{\displaystyle \sum _{i=1}^{\infty }{\frac {1}{i(i+k)}}=\lim _{n\rightarrow \infty }{\frac {1}{k}}\cdot {\Biggr [}H_{k}+H_{n}-H_{n+k}{\Biggr ]}={\frac {1}{k}}\cdot {\Biggr [}\lim _{n\rightarrow \infty }H_{k}+\lim _{n\rightarrow \infty }\left(H_{n}-H_{n+k}\right){\Biggr ]}={\frac {H_{k}}{k}}}"></span></dd></dl> <p>Következtetésképpen megállapíthatjuk, hogy a <span class="texhtml mvar" style="font-style:italic;">k</span> különbségű pozitív téglalapszámok reciprokainak összege <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {H_{k}}{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {H_{k}}{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce90939e2333a212283ea7592c3535ec63edfa8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:3.856ex; height:5.343ex;" alt="{\displaystyle {\frac {H_{k}}{k}}}"></span>, ahol <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8af69c8cae9ee6ba302b6b3f4b0618ac08e427" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.02ex; height:2.509ex;" alt="{\displaystyle H_{k}}"></span> a k-dik <a href="/wiki/Harmonikus_sz%C3%A1m" title="Harmonikus szám">harmonikus szám</a>. </p> <div class="mw-heading mw-heading2"><h2 id="További_tulajdonságaik"><span id="Tov.C3.A1bbi_tulajdons.C3.A1gaik"></span>További tulajdonságaik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=5" title="Szakasz szerkesztése: További tulajdonságaik"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Az <span class="texhtml mvar" style="font-style:italic;">n</span>-edik téglalapszám megegyezik az első <span class="texhtml mvar" style="font-style:italic;">n</span> páros egész szám összegével.<sup id="cite_ref-knorr_2-2" class="reference"><a href="#cite_note-knorr-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Ebből következik az is, hogy az összes téglalapszám <a href="/wiki/P%C3%A1ros_%C3%A9s_p%C3%A1ratlan_sz%C3%A1mok" title="Páros és páratlan számok">páros</a>, és közülük egyedül a 2 <a href="/wiki/Pr%C3%ADmsz%C3%A1m" class="mw-redirect" title="Prímszám">prímszám</a>. Szintén a 2 az egyetlen <a href="/wiki/Fibonacci-sz%C3%A1m" class="mw-redirect" title="Fibonacci-szám">Fibonacci-téglalapszám</a> és az egyetlen téglalap <a href="/wiki/Lucas-sz%C3%A1m" class="mw-redirect" title="Lucas-szám">Lucas-szám</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>A <a href="/wiki/N%C3%A9gyzetes_m%C3%A1trix" title="Négyzetes mátrix">négyzetes mátrix</a> átlón kívüli elemeinek száma mindig téglalapszám.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p>A tény, hogy az egymást követő egészek mindig <a href="/wiki/Relat%C3%ADv_pr%C3%ADmek" title="Relatív prímek">relatív prímek</a>, a téglalapszámok pedig két egymást követő egész szorzatai, néhány új tulajdonsághoz vezetnek. A téglalapszám minden prímtényezője az őt alkotó tényezők közül pontosan az egyikben fordul elő. Tehát egy téglalapszám <a href="/wiki/Csakkor" class="mw-redirect" title="Csakkor">csakkor</a> <a href="/wiki/N%C3%A9gyzetmentes" class="mw-redirect" title="Négyzetmentes">négyzetmentes</a>, ha <span class="texhtml mvar" style="font-style:italic;">n</span> és <span class="texhtml"><i>n</i> + 1</span> is négyzetmentesek. A téglalapszámok különböző prímtényezőinek száma megegyezik az <span class="texhtml mvar" style="font-style:italic;">n</span> és <span class="texhtml"><i>n</i> + 1</span> különböző prímtényezői számának összegével. </p><p>További tulajdonsága a téglalapszámoknak, hogy az <i>n</i>-nél 0,5-del nagyobb szám négyzete pont az n-edik téglalapszámnál 0,25-dal nagyobb. Például: 7,5<sup>2</sup> = 56,25. Ezért az 5-re végződő egész <a href="/wiki/N%C3%A9gyzetsz%C3%A1mok" title="Négyzetszámok">négyzetszámok</a> négyzete 25-re végződik úgy, hogy az azt megelőző számjegyek téglalapszámot alkotnak. </p><p>Még egy másik tulajdonságuk, hogy bármelyik <i>n</i> alapú <a href="/wiki/Sz%C3%A1mrendszer" title="Számrendszer">számrendszerben</a> az n-edik, vagyis a számrendszer <a href="/wiki/Radix" title="Radix">alapszámával</a> megegyező sorszámú téglalapszám <i>110</i> alakban írható fel. Például a <a href="/wiki/Nyolcas_sz%C3%A1mrendszer" title="Nyolcas számrendszer">nyolcas számrendszerben</a> az 110<sub>8</sub> szám <a href="/wiki/72_(sz%C3%A1m)" title="72 (szám)">72</a>-t jelent, amely pont a 8. téglalapszám. A <a href="/wiki/T%C3%ADzes_sz%C3%A1mrendszer" title="Tízes számrendszer">tízes számrendszerben</a> épp a tizedik téglalapszám írható fel <a href="/wiki/110_(sz%C3%A1m)" title="110 (szám)">110 (száztíz)</a> alakban. Ennek oka ugyanaz, ami miatt a <span class="texhtml"><i>n</i><sup>2</sup> + <i>n</i></span> is az egyik kiszámítási képlet alternatívája, vagyis az n-edik téglalapszám az <i>n</i> szám (számrendszer alapszáma) első két <a href="/wiki/Hatv%C3%A1ny" title="Hatvány">hatványának</a> összege. </p> <div class="mw-heading mw-heading2"><h2 id="További_információk"><span id="Tov.C3.A1bbi_inform.C3.A1ci.C3.B3k"></span>További információk</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=6" title="Szakasz szerkesztése: További információk"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://refkol.ro/tuzson/cikkeim/a%20figuralis%20szamokrol2.pdf">Tuzson Zoltán: A figurális számokról (II)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Fordítás"><span id="Ford.C3.ADt.C3.A1s"></span>Fordítás</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=7" title="Szakasz szerkesztése: Fordítás"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Ez a szócikk részben vagy egészben a <i><a href="https://en.wikipedia.org/wiki/Pronic_number" class="extiw" title="en:Pronic number">Pronic number</a></i> című angol Wikipédia-szócikk <span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/wiki/Pronic_number?oldid=692322416">ezen változatának</a> fordításán alapul.</span> Az eredeti cikk szerkesztőit annak laptörténete sorolja fel. Ez a jelzés csupán a megfogalmazás eredetét és a szerzői jogokat jelzi, nem szolgál a cikkben szereplő információk forrásmegjelöléseként.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Jegyzetek">Jegyzetek</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=T%C3%A9glalapsz%C3%A1mok&action=edit&section=8" title="Szakasz szerkesztése: Jegyzetek"><span>szerkesztés</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="ref-1col"> <ol class="references"> <li id="cite_note-bon-1"><span class="mw-cite-backlink">↑ <a href="#cite_ref-bon_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-bon_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=The+Book+of+Numbers&rft.aulast=Conway&rft.aufirst=J.+H.&rft.date=1996&rft.pages=Figure+2.15%2C+p.%26nbsp%3B34&rft.place=New+York&rft.place=New+York&rft.pub=Copernicus"><cite id="CITEREFConwayGuy1996"><a href="/w/index.php?title=John_H._Conway&action=edit&redlink=1" class="new" title="John H. Conway (a lap nem létezik)">Conway, J. H.</a> & <a href="/w/index.php?title=Richard_K._Guy&action=edit&redlink=1" class="new" title="Richard K. Guy (a lap nem létezik)">Guy, R. K.</a> (1996), <i>The Book of Numbers</i>, New York: Copernicus, Figure 2.15, p. 34</cite></span>.</span> </li> <li id="cite_note-knorr-2"><span class="mw-cite-backlink">↑ <a href="#cite_ref-knorr_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-knorr_2-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-knorr_2-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=The+evolution+of+the+Euclidean+elements&rft.aulast=Knorr&rft.aufirst=Wilbur+Richard&rft.date=1975&rft.pages=pp.+144%E2%80%93150&rft.place=Dordrecht-Boston%2C+Mass.&rft.place=Dordrecht-Boston%2C+Mass.&rft.pub=D.+Reidel+Publishing+Co.&rft.isbn=90-277-0509-7&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D_1H6BwAAQBAJ%26pg%3DPA144"><cite id="CITEREFKnorr1975"><a href="/w/index.php?title=Wilbur_Knorr&action=edit&redlink=1" class="new" title="Wilbur Knorr (a lap nem létezik)">Knorr, Wilbur Richard</a> (1975), <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=_1H6BwAAQBAJ&pg=PA144">The evolution of the Euclidean elements</a></i>, Dordrecht-Boston, Mass.: D. Reidel Publishing Co., pp. 144–150, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/90-277-0509-7" title="Speciális:Könyvforrások/90-277-0509-7">ISBN 90-277-0509-7</a><span class="printonly">, <<a rel="nofollow" class="external free" href="https://books.google.com/books?id=_1H6BwAAQBAJ&pg=PA144">https://books.google.com/books?id=_1H6BwAAQBAJ&pg=PA144</a>></span></cite></span>.</span> </li> <li id="cite_note-hist-3"><span class="mw-cite-backlink"><a href="#cite_ref-hist_3-0">↑</a></span> <span class="reference-text"><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=Historical+Encyclopedia+of+Natural+and+Mathematical+Sciences%2C+Volume+1&rft.aulast=Ben-Menahem&rft.aufirst=Ari&rft.date=2009&rft.pages=p.+161&rft.pub=Springer-Verlag&rft.isbn=9783540688310&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D9tUrarQYhKMC%26pg%3DPA161"><cite id="CITEREFBen-Menahem2009">Ben-Menahem, Ari (2009), <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=9tUrarQYhKMC&pg=PA161">Historical Encyclopedia of Natural and Mathematical Sciences, Volume 1</a></i>, Springer reference, Springer-Verlag, p. 161, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9783540688310" title="Speciális:Könyvforrások/9783540688310">ISBN 9783540688310</a><span class="printonly">, <<a rel="nofollow" class="external free" href="https://books.google.com/books?id=9tUrarQYhKMC&pg=PA161">https://books.google.com/books?id=9tUrarQYhKMC&pg=PA161</a>></span></cite></span>.</span> </li> <li id="cite_note-telescope-4"><span class="mw-cite-backlink"><a href="#cite_ref-telescope_4-0">↑</a></span> <span class="reference-text">Marc Frantz: <a rel="nofollow" class="external text" href="https://books.google.hu/books?id=SHJ39945R1kC&pg=PA467&redir_esc=y#v=onepage&q&f=false">The Telescoping Series in Perspective</a>. In Caren L. Diefenderfer – Roger B. Nelsen: <i>The Calculus Collection: A Resource for AP and Beyond.</i> (angolul) <a href="/wiki/Washington_(f%C5%91v%C3%A1ros)" title="Washington (főváros)">Washington, D.C.</a>: Mathematical Association of America. 2009.  467–468. o. = Classroom Resource Materials, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9780883857618" title="Speciális:Könyvforrások/9780883857618">ISBN 9780883857618</a> Hozzáférés: 2018. május 3. <small><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fhu.wikipedia.org%3AT%C3%A9glalapsz%C3%A1mok&rft.atitle=The+Telescoping+Series+in+Perspective&rft.au=Caren+L.+Diefenderfer&rft.au=Roger+B.+Nelsen&rft.btitle=The+Calculus+Collection%3A+A+Resource+for+AP+and+Beyond&rft.date=2009&rft.genre=book&rft.isbn=9780883857618&rft.pages=467%E2%80%93468&rft.place=Washington%2C+D.C.&rft.pub=Mathematical+Association+of+America&rft.series=Classroom+Resource+Materials&rft_id=https%3A%2F%2Fbooks.google.hu%2Fbooks%3Fid%3DSHJ39945R1kC%26pg%3DPA467%26redir_esc%3Dy%23v%3Donepage%26q%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></small>.</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text"><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=Pronic+Lucas+numbers&rft.aulast=McDaniel&rft.aufirst=Wayne+L.&rft.date=1998&rft.volume=36&rft.issue=1&rft.pages=60%E2%80%9362&rft_id=http%3A%2F%2Fwww.mathstat.dal.ca%2FFQ%2FScanned%2F36-1%2Fmcdaniel2.pdf"><cite id="CITEREFMcDaniel1998">McDaniel, Wayne L. (1998), "<a rel="nofollow" class="external text" href="http://www.mathstat.dal.ca/FQ/Scanned/36-1/mcdaniel2.pdf">Pronic Lucas numbers</a>", <i><a href="/w/index.php?title=Fibonacci_Quarterly&action=edit&redlink=1" class="new" title="Fibonacci Quarterly (a lap nem létezik)">Fibonacci Quarterly</a></i> <b>36</b> (1): 60–62<span class="printonly">, <<a rel="nofollow" class="external free" href="http://www.mathstat.dal.ca/FQ/Scanned/36-1/mcdaniel2.pdf">http://www.mathstat.dal.ca/FQ/Scanned/36-1/mcdaniel2.pdf</a>></span>. Hozzáférés ideje: 2016-02-05</cite></span>.</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=Pronic+Fibonacci+numbers&rft.aulast=McDaniel&rft.aufirst=Wayne+L.&rft.date=1998&rft.volume=36&rft.issue=1&rft.pages=56%E2%80%9359&rft_id=http%3A%2F%2Fwww.fq.math.ca%2FScanned%2F36-1%2Fmcdaniel1.pdf"><cite id="CITEREFMcDaniel1998">McDaniel, Wayne L. (1998), "<a rel="nofollow" class="external text" href="http://www.fq.math.ca/Scanned/36-1/mcdaniel1.pdf">Pronic Fibonacci numbers</a>", <i><a href="/w/index.php?title=Fibonacci_Quarterly&action=edit&redlink=1" class="new" title="Fibonacci Quarterly (a lap nem létezik)">Fibonacci Quarterly</a></i> <b>36</b> (1): 56–59<span class="printonly">, <<a rel="nofollow" class="external free" href="http://www.fq.math.ca/Scanned/36-1/mcdaniel1.pdf">http://www.fq.math.ca/Scanned/36-1/mcdaniel1.pdf</a>></span></cite></span>.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><a href="#cite_ref-7">↑</a></span> <span class="reference-text"><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=Applied+Factor+Analysis&rft.aulast=Rummel&rft.aufirst=Rudolf+J.&rft.date=1988&rft.pages=p.+319&rft.pub=Northwestern+University+Press&rft.isbn=9780810108240&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dg_eNa_XzyEIC%26pg%3DPA319"><cite id="CITEREFRummel1988">Rummel, Rudolf J. (1988), <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=g_eNa_XzyEIC&pg=PA319">Applied Factor Analysis</a></i>, Northwestern University Press, p. 319, <a href="/wiki/Speci%C3%A1lis:K%C3%B6nyvforr%C3%A1sok/9780810108240" title="Speciális:Könyvforrások/9780810108240">ISBN 9780810108240</a><span class="printonly">, <<a rel="nofollow" class="external free" href="https://books.google.com/books?id=g_eNa_XzyEIC&pg=PA319">https://books.google.com/books?id=g_eNa_XzyEIC&pg=PA319</a>></span></cite></span>.</span> </li> </ol> </div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r26593303">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline 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href="/wiki/Pr%C3%ADmfelbont%C3%A1s" title="Prímfelbontás">Prímfelbontás</a></li> <li><a href="/wiki/Oszthat%C3%B3s%C3%A1g" title="Oszthatóság">Osztó</a></li> <li><a href="/w/index.php?title=Unitary_divisor&action=edit&redlink=1" class="new" title="Unitary divisor (a lap nem létezik)">Unitary divisor</a></li> <li><a href="/wiki/Oszt%C3%B3sz%C3%A1m-f%C3%BCggv%C3%A9ny" title="Osztószám-függvény">Osztószám-függvény</a></li> <li><a href="/wiki/Oszt%C3%B3%C3%B6sszeg-f%C3%BCggv%C3%A9ny" title="Osztóösszeg-függvény">Osztóösszeg-függvényregul</a></li> <li><a href="/wiki/Pr%C3%ADmt%C3%A9nyez%C5%91" title="Prímtényező">Prímtényező</a></li> <li><a href="/wiki/A_sz%C3%A1melm%C3%A9let_alapt%C3%A9tele" title="A számelmélet alaptétele">A számelmélet alaptétele</a></li> <li><a href="/wiki/Aritmetikus_sz%C3%A1mok" title="Aritmetikus számok">Aritmetikus számok</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><span typeof="mw:File"><a href="/wiki/F%C3%A1jl:Lattice_of_the_divisibility_of_60.svg" class="mw-file-description" title="60 osztói"><img alt="60 osztói" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/200px-Lattice_of_the_divisibility_of_60.svg.png" decoding="async" width="200" height="160" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/300px-Lattice_of_the_divisibility_of_60.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Lattice_of_the_divisibility_of_60.svg/400px-Lattice_of_the_divisibility_of_60.svg.png 2x" data-file-width="313" data-file-height="250" /></a></span></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Prímtényezős felbontás</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Pr%C3%ADmsz%C3%A1mok" title="Prímszámok">Prím</a></li> <li><a href="/wiki/%C3%96sszetett_sz%C3%A1mok" title="Összetett számok">Összetett</a></li> <li><a href="/wiki/F%C3%A9lpr%C3%ADmek" title="Félprímek">Félprím</a></li> <li><a class="mw-selflink selflink">Téglalapszám</a></li> <li><a href="/wiki/Szfenikus_sz%C3%A1mok" title="Szfenikus számok">Szfenikus</a></li> <li><a href="/wiki/N%C3%A9gyzetmentes_sz%C3%A1m" title="Négyzetmentes szám">Négyzetmentes</a></li> <li><a href="/wiki/N%C3%A9gyzetteljes_sz%C3%A1mok" title="Négyzetteljes számok">Hatványteljes</a></li> <li><a href="/wiki/Teljes_hatv%C3%A1ny" title="Teljes hatvány">Teljes hatvány</a></li> <li><a href="/wiki/Achilles-sz%C3%A1mok" title="Achilles-számok">Achilles</a></li> <li><a href="/wiki/Sima_sz%C3%A1mok" title="Sima számok">Sima</a></li> <li><a href="/wiki/Szab%C3%A1lyos_sz%C3%A1mok" title="Szabályos számok">Szabályos</a></li> <li><a href="/wiki/Durva_sz%C3%A1mok" title="Durva számok">Durva</a></li> <li><a href="/wiki/Szokatlan_sz%C3%A1mok" title="Szokatlan 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lap nem létezik)">Friendly</a></li> <li><a href="/w/index.php?title=Friendly_number&action=edit&redlink=1" class="new" title="Friendly number (a lap nem létezik)">Solitary</a></li> <li><a href="/w/index.php?title=Sublime_number&action=edit&redlink=1" class="new" title="Sublime number (a lap nem létezik)">Sublime</a></li> <li><a href="/wiki/Oszt%C3%B3harmonikus_sz%C3%A1mok" title="Osztóharmonikus számok">Osztóharmonikus</a></li> <li><a href="/w/index.php?title=Frugal_number&action=edit&redlink=1" class="new" title="Frugal number (a lap nem létezik)">Frugal</a></li> <li><a href="/w/index.php?title=Equidigital_number&action=edit&redlink=1" class="new" title="Equidigital number (a lap nem létezik)">Equidigital</a></li> <li><a href="/w/index.php?title=Extravag%C3%A1ns_sz%C3%A1mok&action=edit&redlink=1" class="new" title="Extravagáns számok (a lap nem létezik)">Extravagáns</a></li></ul> </div></td></tr></tbody></table></div></div><noscript><img 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Téglalapszámok – Wikipédia