{"id":755,"date":"2026-10-03T15:24:02","date_gmt":"2026-10-03T14:24:02","guid":{"rendered":"https:\/\/wpcalc.com\/tr\/ardisik-karelerin-toplami\/"},"modified":"2026-10-03T15:24:02","modified_gmt":"2026-10-03T14:24:02","slug":"ardisik-karelerin-toplami","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/tr\/matematik\/ardisik-karelerin-toplami\/","title":{"rendered":"Ard\u0131\u015f\u0131k Karelerin Toplam\u0131 Hesaplay\u0131c\u0131"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 id=\"calculator-title\" class=\"text-lg font-display font-bold\">Ard\u0131\u015f\u0131k Kareleri Toplay\u0131n<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"755\" data-favorite-title=\"Ard\u0131\u015f\u0131k Karelerin Toplam\u0131\" data-favorite-url=\"https:\/\/wpcalc.com\/tr\/matematik\/ardisik-karelerin-toplami\/\" data-favorite-excerpt=\"\u0130lk n do\u011fal say\u0131n\u0131n karelerinin toplam\u0131n\u0131 hesaplay\u0131n: 1\u00b2 + 2\u00b2 + ... + n\u00b2. Terim say\u0131s\u0131n\u0131 girin ve n(n+1)(2n+1)\/6 form\u00fcl\u00fcyle an\u0131nda sonucu al\u0131n.\" aria-label=\"Favorilere ekle\" data-favorite-icon=\"icon icon-arithmetic\">\n    <span class=\"icon icon-shape-star-empty\" aria-hidden=\"true\"><\/span>\n  <\/button>\n  <div class=\"calcdash-favorite__tip\" aria-hidden=\"true\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n  <span class=\"sr-only\" role=\"status\" id=\"favorite-status\"><\/span>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/tr\/matematik\/ardisik-karelerin-toplami\/\" method=\"POST\" class=\"calculator\" id=\"calculator-755\" data-post=\"755\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Giri\u015f alanlar\u0131<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Terim say\u0131s\u0131 (n)<\/label><input type=\"text\" name=\"terim_says_n\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"5\" min=\"1\" value=\"5\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Yeniden hesapla\" id=\"calculate-button\" data-post=\"755\">Hesapla<\/button><button type=\"reset\">S\u0131f\u0131rla<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Otomatik hesapla<\/label><small>Etkinle\u015ftirilirse, herhangi bir de\u011feri de\u011fi\u015ftirdi\u011finizde sonu\u00e7 otomatik olarak g\u00fcncellenir.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Sonu\u00e7lar\u0131n\u0131z<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Karelerin toplam\u0131<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Sonucu kopyala\" aria-label=\"Sonucu kopyala\"><span class=\"copy-icon icon icon-document-copy\" aria-hidden=\"true\"><\/span><\/button><\/div><\/div><div class=\"result-actions\"><button type=\"button\" class=\"copy-link\"><svg viewBox=\"0 0 20 20\" fill=\"currentColor\" aria-hidden=\"true\"><path d=\"M12.232 4.232a2.5 2.5 0 0 1 3.536 3.536l-1.225 1.224a.75.75 0 0 0 1.061 1.06l1.224-1.224a4 4 0 0 0-5.656-5.656l-3 3a4 4 0 0 0 .225 5.865.75.75 0 0 0 .977-1.138 2.5 2.5 0 0 1-.142-3.667l3-3Z\"\/><path d=\"M11.603 7.963a.75.75 0 0 0-.977 1.138 2.5 2.5 0 0 1 .142 3.667l-3 3a2.5 2.5 0 0 1-3.536-3.536l1.225-1.224a.75.75 0 0 0-1.061-1.06l-1.224 1.224a4 4 0 1 0 5.656 5.656l3-3a4 4 0 0 0-.225-5.865Z\"\/><\/svg><span class=\"copy-link__text\">Bu hesaplaman\u0131n ba\u011flant\u0131s\u0131n\u0131 kopyala<\/span><\/button><span class=\"sr-only\" role=\"status\" id=\"copy-link-status\"><\/span><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Yorum b\u0131rak<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"755\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">Bu sonu\u00e7 hakk\u0131nda ne d\u00fc\u015f\u00fcn\u00fcyorsunuz?<\/p><div role=\"group\" aria-labelledby=\"calc-reactions-title\" class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Be\u011fendim <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Faydal\u0131 <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Kafam kar\u0131\u015ft\u0131 <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Hayal k\u0131r\u0131kl\u0131\u011f\u0131 <\/span><\/button><button type=\"button\" aria-pressed=\"false\" class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\" aria-hidden=\"true\">\u274c <\/span><span class=\"reaction-description font-medium\">Hatal\u0131 <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><div class=\"calcdash-ad not-prose calcdash-ad--calc_result calcdash-ad--in-article\"><ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-1721569815777345\" data-ad-slot=\"6317458308\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\"><\/ins><script>(adsbygoogle = window.adsbygoogle || []).push({});<\/script><\/div>\n<style id=\"wpcoderpro-inline-1\">.formula-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Ard\u0131\u015f\u0131k Karelerin Toplam\u0131 Form\u00fcl\u00fc<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Form\u00fcl<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n\\[ 1^2 + 2^2 + 3^2 + \\dots + n^2 = \\frac{n(n + 1)(2n + 1)}{6} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Burada:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 terim say\u0131s\u0131, yani karesi al\u0131nan son do\u011fal say\u0131<\/li>\n<\/ul>\n<p>Form\u00fcl, ilk \\(n\\) pozitif tam say\u0131n\u0131n karelerinin toplam\u0131n\u0131 tek ad\u0131mda, her terimi tek tek kare al\u0131p toplamaya gerek kalmadan tam olarak verir.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-2\">.calculation-box {\r\n\tborder-bottom: none !important;\r\n}<\/style><section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Ard\u0131\u015f\u0131k Karelerin Toplam\u0131 \u2013 Hesaplama \u00d6rne\u011fi<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>\u00d6rnek:<\/strong> ilk 5 do\u011fal say\u0131n\u0131n karelerinin toplam\u0131n\u0131 bulun.<\/p>\n<p>\\[ 1^2 + 2^2 + 3^2 + 4^2 + 5^2 = \\frac{5 \\cdot 6 \\cdot 11}{6} = \\frac{330}{6} = 55 \\]<\/p>\n<p><strong>Sonu\u00e7:<\/strong> \\(n = 5\\) i\u00e7in ard\u0131\u015f\u0131k karelerin toplam\u0131 55'tir.<\/p>\n<p>\n\r\n<\/section>\n<h2>Hesaplay\u0131c\u0131 Nas\u0131l Kullan\u0131l\u0131r?<\/h2>\n<ol>\n<li><strong>Terim say\u0131s\u0131 (n)<\/strong> alan\u0131na pozitif bir tam say\u0131 girin \u2014 \u00f6rne\u011fin 5.<\/li>\n<li>Hesaplay\u0131c\u0131 <strong>Karelerin toplam\u0131n\u0131<\/strong> \u2014 \\(1^2 + 2^2 + \\dots + n^2\\) de\u011ferini \u2014 an\u0131nda g\u00f6r\u00fcnt\u00fcler.<\/li>\n<li>\\(n\\) de\u011ferini de\u011fi\u015ftirerek toplam\u0131 farkl\u0131 bir terim say\u0131s\u0131 i\u00e7in yeniden hesaplay\u0131n.<\/li>\n<\/ol>\n<h2>Ard\u0131\u015f\u0131k Karelerin Toplam\u0131 Nedir?<\/h2>\n<p>Ard\u0131\u015f\u0131k karelerin toplam\u0131, ilk \\(n\\) do\u011fal say\u0131n\u0131n her birinin karesini al\u0131p sonu\u00e7lar\u0131 birlikte ekledi\u011finizde elde etti\u011finiz de\u011ferdir: \\(1^2 + 2^2 + 3^2 + \\dots + n^2\\). Bunlar\u0131 elle toplamak h\u0131zla yorucu hale gelir \u2014 \\(n = 100\\) i\u00e7in 100 kare hesaplay\u0131p toplaman\u0131z gerekirdi. Kapal\u0131 form\u00fcl \\(\\frac{n(n+1)(2n+1)}{6}\\) t\u00fcm g\u00f6revi \u00fc\u00e7 \u00e7arpma, bir toplama ve bir b\u00f6lme i\u015flemine indirir ve \\(n\\) ne kadar b\u00fcy\u00fck olursa olsun tam cevab\u0131 verir.<\/p>\n<p>Bu, matemati\u011fin en eski sonu\u00e7lar\u0131ndan biridir. Ar\u015fimet iki bin y\u0131ldan fazla \u00f6nce e\u015fde\u011fer bir ba\u011f\u0131nt\u0131 kullanm\u0131\u015ft\u0131r ve matematiksel t\u00fcmevar\u0131mla yap\u0131lan titiz bir kan\u0131t bug\u00fcn cebir derslerinde klasik bir al\u0131\u015ft\u0131rmad\u0131r. Form\u00fcl bir \"kuvvet toplamlar\u0131\" form\u00fclleri ailesine aittir; k\u00fcpler i\u00e7in analo\u011fik sonu\u00e7 daha da basittir: ilk \\(n\\) k\u00fcp\u00fcn toplam\u0131 \\(\\left(\\frac{n(n+1)}{2}\\right)^2\\)'ye e\u015fittir; bunu <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/ardisik-kuplerin-toplami\/\">Ard\u0131\u015f\u0131k K\u00fcplerin Toplam\u0131 Hesaplay\u0131c\u0131<\/a> ile do\u011frulayabilirsiniz.<\/p>\n<p>S\u0131n\u0131f\u0131n \u00f6tesinde, kare toplamlar\u0131 istatistik ve veri analizinde kar\u015f\u0131m\u0131za \u00e7\u0131kar. \u00d6rne\u011fin \\(1, 2, \\dots, n\\) say\u0131lar\u0131n\u0131n karesel ortalamas\u0131 (RMS) do\u011frudan bu toplamdan olu\u015fturulur \u2014 bkz. <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/karesel-ortalama-hesaplama\/\">Karesel Ortalama Hesaplay\u0131c\u0131<\/a>. \u0130statistikte varyans\u0131n hesaplanmas\u0131 kare sapmalar\u0131n toplanmas\u0131n\u0131 gerektirir; bilgisayar biliminde ise form\u00fcl, i\u00e7 i\u00e7e d\u00f6ng\u00fclerin zaman karma\u015f\u0131kl\u0131\u011f\u0131n\u0131 analiz etmek i\u00e7in kullan\u0131l\u0131r: \\(1, 2, \\dots, n\\) kez \u00e7al\u0131\u015fan bir i\u00e7 d\u00f6ng\u00fc, \u00e7ift i\u00e7in tam olarak \\(\\frac{n(n+1)}{2}\\) yineleme verir \u2014 ve kare versiyonlar\u0131, kabarc\u0131k s\u0131ralamas\u0131 gibi algoritmalarda kar\u015f\u0131la\u015ft\u0131rmalar\u0131n takaslarla birle\u015fiminde ortaya \u00e7\u0131kar.<\/p>\n<p>Sonu\u00e7, b\u00fcy\u00fck \\(n\\) i\u00e7in kabaca \\(\\frac{n^3}{3}\\) gibi b\u00fcy\u00fcr. Her terim pozitif oldu\u011fundan toplam \\(n\\) ile birlikte her zaman artar ve her zaman bir tam say\u0131d\u0131r \u2014 \\(n(n+1)(2n+1)\\) \u00e7arp\u0131m\u0131n\u0131n 6 ile b\u00f6l\u00fcnebilirli\u011fi garanti edildir, dolay\u0131s\u0131yla b\u00f6lme asla kesir \u00fcretmez. Bu, ara \u00e7arp\u0131m tam say\u0131 t\u00fcr\u00fcn\u00fc a\u015fmad\u0131\u011f\u0131 s\u00fcrece form\u00fcl\u00fcn programlamada tam say\u0131 aritmeti\u011fiyle g\u00fcvenle kullan\u0131labilmesini sa\u011flar.<\/p>\n<h2>Varsay\u0131mlar, S\u0131n\u0131rlamalar ve \u0130pu\u00e7lar\u0131<\/h2>\n<ul>\n<li><strong>Yaln\u0131zca pozitif tam say\u0131lar.<\/strong> Hesaplay\u0131c\u0131 \\(n \\geq 1\\) de\u011ferlerini kabul eder. Kareleri rastgele bir say\u0131dan ba\u015flayarak veya \u00f6zel bir ad\u0131mla toplamaz \u2014 her zaman \\(1^2\\)'den ba\u015flar. \\(k^2 + (k+1)^2 + \\dots + n^2\\) toplam\u0131n\u0131 hesaplamak i\u00e7in \\(n\\) i\u00e7in toplam\u0131 hesaplay\u0131p \\(k-1\\) i\u00e7in olan toplam\u0131 \u00e7\u0131kar\u0131n.<\/li>\n<li><strong>\\(n\\)'in b\u00fcy\u00fck de\u011ferleri.<\/strong> Sonu\u00e7 k\u00fcbik b\u00fcy\u00fcd\u00fc\u011f\u00fcnden say\u0131lar h\u0131zla b\u00fcy\u00fcr: \\(n = 1000\\) i\u00e7in toplam \u00e7oktan 338.350'dir. Hesaplay\u0131c\u0131 b\u00fcy\u00fck girdileri i\u015fler, ancak form\u00fcl\u00fc kendiniz uygularsan\u0131z ta\u015fmay\u0131 \u00f6nlemek i\u00e7in 64 bit tam say\u0131 veya kayan nokta t\u00fcr\u00fc kullan\u0131n.<\/li>\n<li><strong>Do\u011frulama kontrol\u00fc.<\/strong> Bir sonucu h\u0131zl\u0131ca do\u011frulaman\u0131n yolu: ilk \\(n\\) karenin toplam\u0131 her zaman \\(\\frac{n^3}{3}\\) ile \\(\\frac{(n+1)^3}{3}\\) aras\u0131ndad\u0131r; bu, yaz\u0131m hatalar\u0131n\u0131 veya ta\u015fma hatalar\u0131n\u0131 yakalamaya yard\u0131mc\u0131 olur.<\/li>\n<li><strong>\u0130lgili toplamlar.<\/strong> \u0130lk \\(n\\) tam say\u0131n\u0131n d\u00fcz toplam\u0131na ihtiyac\u0131n\u0131z varsa <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/aritmetik-dizi\/\">Aritmetik Dizi Hesaplay\u0131c\u0131<\/a>'y\u0131 kullan\u0131n; say\u0131 sat\u0131rlar\u0131ndan olu\u015fan bir toplam i\u00e7in <a href=\"https:\/\/wpcalc.com\/tr\/matematik\/floyd-ucgeni-toplami\/\">Floyd \u00dc\u00e7geni Toplam\u0131 Hesaplay\u0131c\u0131<\/a>'y\u0131 deneyin.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>\u0130lk n do\u011fal say\u0131n\u0131n karelerinin toplam\u0131n\u0131 hesaplay\u0131n: 1\u00b2 + 2\u00b2 + ... + n\u00b2. Terim say\u0131s\u0131n\u0131 girin ve n(n+1)(2n+1)\/6 form\u00fcl\u00fcyle an\u0131nda sonucu al\u0131n.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[8],"tags":[18],"class_list":["post-755","post","type-post","status-publish","format-standard","hentry","category-matematik","tag-aritmetik-hesap-makineleri"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/755","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/comments?post=755"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/755\/revisions"}],"predecessor-version":[{"id":3213,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/posts\/755\/revisions\/3213"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/media?parent=755"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/categories?post=755"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/tr\/wp-json\/wp\/v2\/tags?post=755"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}