{"id":61472,"date":"2021-02-02T16:18:16","date_gmt":"2021-02-02T14:18:16","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61472"},"modified":"2026-09-30T20:50:10","modified_gmt":"2026-09-30T17:50:10","slug":"sum-consecutive-cubes","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-cubes\/","title":{"rendered":"Sum of Consecutive Cubes 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 Cubes<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61472\" data-favorite-title=\"Sum of Consecutive Cubes\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-cubes\/\" data-favorite-excerpt=\"Calculate the sum of the first n consecutive cubes instantly. Enter the number of terms and get the result using the closed-form formula (n(n+1)\/2)^2, with a worked example and explanation.\" 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-cubes\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61472\" data-post=\"61472\"><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=\"61472\">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 cubes<\/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\" 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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 Cubes 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\\[ \\sum_{k=1}^{n} k^3 = \\left( \\frac{n(n+1)}{2} \\right)^2 \\]\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 (a positive integer)<\/li>\n<li>\\(k^3\\) \u2014 the cube of each term, i.e. \\(k \\times k \\times k\\)<\/li>\n<\/ul>\n<p>This identity, known as Nicomachus's theorem, says that the sum of the cubes of the first \\(n\\) natural numbers equals the square of the sum of those same numbers. It replaces repeated multiplication and addition with a single closed-form expression.<\/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 Cubes \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the sum of the cubes of the first 5 natural numbers, \\(1^3 + 2^3 + 3^3 + 4^3 + 5^3\\).<\/p>\n<p>\\[ \\sum_{k=1}^{5} k^3 = \\left( \\frac{5 \\times (5+1)}{2} \\right)^2 = \\left( \\frac{30}{2} \\right)^2 = 15^2 = 225 \\]<\/p>\n<p>Check by direct addition: \\(1 + 8 + 27 + 64 + 125 = 225\\).<\/p>\n<p><strong>Result:<\/strong> the sum of the first 5 consecutive cubes is 225.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to use the calculator<\/h2>\n<ol>\n<li>Enter a positive whole number in the <strong>Number of terms (n)<\/strong> field \u2014 for example, 5 to sum the cubes from \\(1^3\\) to \\(5^3\\).<\/li>\n<li>The calculator instantly displays the answer in the <strong>Sum of cubes<\/strong> field.<\/li>\n<li>Change the value of \\(n\\) at any time to compute a different sum; the result updates automatically.<\/li>\n<\/ol>\n<h2>What is the sum of consecutive cubes?<\/h2>\n<p>The sum of consecutive cubes is the total you get when you add \\(1^3 + 2^3 + 3^3 + \\dots + n^3\\). Each term is a perfect cube, and the series grows quickly: the first few sums are 1, 9, 36, 100, 225, 441\u2026 Computing them by hand is tedious for large \\(n\\), because you have to cube every integer from 1 to \\(n\\) and then add all the results together.<\/p>\n<p>The closed-form formula eliminates that work entirely. It relies on a remarkable fact: the sum of the first \\(n\\) cubes is exactly the square of the \\(n\\)-th triangular number. Since \\(\\frac{n(n+1)}{2}\\) is the familiar formula for \\(1 + 2 + \\dots + n\\), the sum of cubes is simply that quantity squared. For example, \\(1 + 2 + 3 + 4 + 5 = 15\\), and \\(15^2 = 225\\) \u2014 precisely \\(1^3 + 2^3 + 3^3 + 4^3 + 5^3\\).<\/p>\n<p>This result appears often in algebra, number theory, and discrete mathematics. It is a classic example of a power sum identity, alongside the formula for the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-squares\/\">sum of consecutive squares<\/a>, which grows on the order of \\(n^3\\) while the sum of cubes grows on the order of \\(n^4\\). The identity is also closely related to patterns in figurate numbers and appears in the row sums of <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/\">Floyd's triangle<\/a>.<\/p>\n<h2>Interpreting the result<\/h2>\n<p>The number returned by the calculator is always a perfect square \u2014 specifically, the square of the triangular number \\(T_n = \\frac{n(n+1)}{2}\\). This gives you a quick sanity check: if \\(T_n\\) is an integer (it always is for positive integer \\(n\\)), the sum of cubes is \\(T_n^2\\). A related visual interpretation: the sum of the first \\(n\\) cubes counts the total number of cells in a square arrangement built from successively larger square blocks, which is one way to prove the identity geometrically.<\/p>\n<p>Because the result grows as \\(n^4\/4\\) for large \\(n\\), even modest inputs produce large outputs. For \\(n = 100\\), the sum is \\(5050^2 = 25{,}502{,}500\\). The formula handles such cases instantly, whereas direct addition of 100 cubes would take considerably longer.<\/p>\n<h2>Assumptions and limitations<\/h2>\n<ul>\n<li><strong>Starts at 1:<\/strong> the calculator sums cubes beginning from \\(1^3\\). It does not compute sums over an arbitrary range such as \\(5^3 + \\dots + 10^3\\); for that, subtract the result for \\(n = 4\\) from the result for \\(n = 10\\).<\/li>\n<li><strong>Positive integers only:<\/strong> \\(n\\) must be a whole number of at least 1. Fractional or negative inputs do not correspond to a count of terms.<\/li>\n<li><strong>Large values:<\/strong> the sum grows very fast (fourth power of \\(n\\)), so very large inputs may exceed the precision of standard floating-point arithmetic. For everyday values of \\(n\\), results are exact.<\/li>\n<\/ul>\n<p>If you are working with series more generally, you may also find the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/arithmetic-progression\/\">arithmetic progression calculator<\/a> useful for sums that grow linearly rather than as powers.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the sum of the first n consecutive cubes instantly. Enter the number of terms and get the result using the closed-form formula (n(n+1)\/2)^2, with a worked example and explanation.<\/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-61472","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\/61472","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=61472"}],"version-history":[{"count":3,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61472\/revisions"}],"predecessor-version":[{"id":66292,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61472\/revisions\/66292"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61472"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61472"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61472"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}