{"id":61467,"date":"2021-02-02T16:10:19","date_gmt":"2021-02-02T14:10:19","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61467"},"modified":"2026-09-30T20:50:19","modified_gmt":"2026-09-30T17:50:19","slug":"sum-consecutive-squares","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-squares\/","title":{"rendered":"Sum of Consecutive Squares Calculator"},"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\">Add Consecutive Squares<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61467\" data-favorite-title=\"Sum of Consecutive Squares\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-squares\/\" data-favorite-excerpt=\"Calculate the sum of the squares of the first n natural numbers: 1\u00b2 + 2\u00b2 + ... + n\u00b2. Enter the number of terms and get the instant result using the formula n(n+1)(2n+1)\/6.\" aria-label=\"Add to Favorites\" 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\/en\/mathematics\/sum-consecutive-squares\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61467\" data-post=\"61467\"><fieldset class=\"fieldset-input\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field is-single\" id=\"input-1\"><label for=\"field-1\">Number of terms (n)<\/label><input type=\"text\" name=\"number_of_terms_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=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61467\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/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> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Sum of squares<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button type=\"button\" class=\"copy-result\" data-tooltip=\"Copy Result\" aria-label=\"Copy Result\"><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\">Copy link to this calculation<\/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>Leave a Comment<\/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=\"61467\" aria-live=\"polite\" aria-labelledby=\"calc-reactions-title\"><p id=\"calc-reactions-title\" class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/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\">Like <\/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\">Helpful <\/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\">Confused <\/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\">Disappointed <\/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\">Inaccurate <\/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>Sum of Consecutive Squares Formula<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/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>Where:<\/strong><\/p>\n<ul>\n<li>\\(n\\) \u2014 the number of terms, i.e. the last natural number being squared<\/li>\n<\/ul>\n<p>The formula gives the exact sum of the squares of the first \\(n\\) positive integers in a single step, with no need to square and add each term individually.<\/p>\n<p>\r\n<\/section>\n<style id=\"wpcoderpro-inline-459\">.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>Sum of Consecutive Squares \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the sum of the squares of the first 5 natural numbers.<\/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>Result:<\/strong> the sum of consecutive squares for \\(n = 5\\) is 55.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter a positive integer in the <strong>Number of terms (n)<\/strong> field \u2014 for example, 5.<\/li>\n<li>The calculator instantly displays the <strong>Sum of squares<\/strong> \u2014 the value of \\(1^2 + 2^2 + \\dots + n^2\\).<\/li>\n<li>Change the value of \\(n\\) to recalculate the sum for a different number of terms.<\/li>\n<\/ol>\n<h2>What Is the Sum of Consecutive Squares?<\/h2>\n<p>The sum of consecutive squares is the total you get by squaring each of the first \\(n\\) natural numbers and adding the results together: \\(1^2 + 2^2 + 3^2 + \\dots + n^2\\). Adding these by hand quickly becomes tedious \u2014 for \\(n = 100\\), you would have to compute 100 squares and add them up. The closed-form formula \\(\\frac{n(n+1)(2n+1)}{6}\\) reduces the whole task to three multiplications, one addition, and one division, giving the exact answer regardless of how large \\(n\\) is.<\/p>\n<p>This is one of the oldest results in mathematics. Archimedes used an equivalent relation over two thousand years ago, and a rigorous proof by mathematical induction is a classic exercise in algebra courses today. The formula is part of a family of \u201cpower sum\u201d formulas; the analogous result for cubes is even simpler: the sum of the first \\(n\\) cubes equals \\(\\left(\\frac{n(n+1)}{2}\\right)^2\\), which you can verify with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-cubes\/\">Sum of Consecutive Cubes Calculator<\/a>.<\/p>\n<p>Beyond the classroom, sums of squares appear in statistics and data analysis. The root mean square (RMS) of the numbers \\(1, 2, \\dots, n\\), for instance, is built directly from this sum \u2014 see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-mean-calculator-root-mean-square\/\">Quadratic Mean Calculator<\/a>. In statistics, computing variance requires summing squared deviations, and in computer science the formula is used to analyze the time complexity of nested loops, where an inner loop running \\(1, 2, \\dots, n\\) times yields exactly \\(\\frac{n(n+1)}{2}\\) iterations for the pair \u2014 and squared versions appear in algorithms like bubble sort comparisons combined with swaps.<\/p>\n<p>The result grows roughly like \\(\\frac{n^3}{3}\\) for large \\(n\\). Because each term is positive, the sum is always increasing in \\(n\\), and it is always an integer \u2014 the product \\(n(n+1)(2n+1)\\) is guaranteed to be divisible by 6, so the division never produces a fraction. This makes the formula safe to use in programming with integer arithmetic, provided the intermediate product does not overflow the integer type.<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>Positive integers only.<\/strong> The calculator accepts \\(n \\geq 1\\). It does not sum squares starting from an arbitrary number or with a custom step \u2014 it always starts at \\(1^2\\). To sum \\(k^2 + (k+1)^2 + \\dots + n^2\\), compute the sum for \\(n\\) and subtract the sum for \\(k-1\\).<\/li>\n<li><strong>Large values of \\(n\\).<\/strong> The result grows cubically, so the numbers get large fast: for \\(n = 1000\\), the sum is already 338,350. The calculator handles large inputs, but if you implement the formula yourself, use a 64-bit integer or floating-point type to avoid overflow.<\/li>\n<li><strong>Sanity check.<\/strong> A quick way to verify a result: the sum of the first \\(n\\) squares is always between \\(\\frac{n^3}{3}\\) and \\(\\frac{(n+1)^3}{3}\\), which helps catch typos or overflow errors.<\/li>\n<li><strong>Related sums.<\/strong> If you need the sum of the first \\(n\\) plain integers, use the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/arithmetic-progression\/\">Arithmetic Progression Calculator<\/a>; for a sum built from rows of numbers, try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/\">Floyd\u2019s Triangle Sum Calculator<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the sum of the squares of the first n natural numbers: 1\u00b2 + 2\u00b2 + ... + n\u00b2. Enter the number of terms and get the instant result using the formula n(n+1)(2n+1)\/6.<\/p>\n","protected":false},"author":70,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[72],"class_list":["post-61467","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-arithmetic-calculators"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61467","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/70"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=61467"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61467\/revisions"}],"predecessor-version":[{"id":66293,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61467\/revisions\/66293"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61467"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61467"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61467"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}