{"id":61448,"date":"2021-02-02T15:56:34","date_gmt":"2021-02-02T13:56:34","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61448"},"modified":"2026-09-30T20:20:07","modified_gmt":"2026-09-30T17:20:07","slug":"arithmetic-progression","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/arithmetic-progression\/","title":{"rendered":"Arithmetic Progression 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\">Solve Arithmetic Progressions<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61448\" data-favorite-title=\"Arithmetic Progression\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/arithmetic-progression\/\" data-favorite-excerpt=\"Enter the number of terms, the first term, and the common difference to generate the full arithmetic progression and the sum of its first n terms instantly.\" 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\/arithmetic-progression\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61448\" data-post=\"61448\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">n\u1d57\u02b0 term<\/label><div class=\"term\">n<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"n_term\" id=\"field-1\" placeholder=\"6\" step=\"1\" value=\"6\"\/><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">First term<\/label><div class=\"term\">a<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"first_term\" id=\"field-2\" placeholder=\"5\" step=\"1\" value=\"5\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">Common difference<\/label><div class=\"term\">d<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"common_difference\" id=\"field-3\" placeholder=\"2\" step=\"1\" value=\"2\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61448\">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>Arithmetic progression<\/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 class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Sum of first n terms<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/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=\"61448\" 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>Arithmetic Progression 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\\[ \\text{n-th term: } a_n = a_1 + (n - 1)d \\] \\[ \\text{Sum of the first n terms: } S_n = \\frac{n}{2}\\left[2a_1 + (n - 1)d\\right] \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a_1\\) \u2014 the first term of the sequence<\/li>\n<li>\\(d\\) \u2014 the common difference between consecutive terms<\/li>\n<li>\\(n\\) \u2014 the number of terms<\/li>\n<li>\\(a_n\\) \u2014 the n-th term<\/li>\n<li>\\(S_n\\) \u2014 the sum of the first \\(n\\) terms<\/li>\n<\/ul>\n<p>The first formula tells you the value of any term without listing the whole sequence; the second adds all terms at once. The sum formula also equals \\(S_n = \\frac{n}{2}(a_1 + a_n)\\), i.e. the number of terms times the average of the first and last term.<\/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>Arithmetic Progression \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> find the first 6 terms of an arithmetic progression that starts at 5 with a common difference of 2, and their sum.<\/p>\n<p>\\[ a_n = a_1 + (n - 1)d = 5 + (6 - 1) \\cdot 2 = 15 \\]<\/p>\n<p>\\[ S_6 = \\frac{6}{2}\\left[2 \\cdot 5 + (6 - 1) \\cdot 2\\right] = 3 \\cdot 20 = 60 \\]<\/p>\n<p><strong>Result:<\/strong> the progression is 5, 7, 9, 11, 13, 15 and the sum of the first 6 terms is 60.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Calculator<\/h2>\n<ol>\n<li>Enter the <strong>n\u1d57\u02b0 term<\/strong> field \u2014 this is how many terms of the progression you want to generate (e.g. 6).<\/li>\n<li>Enter the <strong>First term<\/strong> \\(a_1\\), the value the sequence starts from (e.g. 5).<\/li>\n<li>Enter the <strong>Common difference<\/strong> \\(d\\), the constant added to each term (e.g. 2). Use a negative value for a decreasing sequence.<\/li>\n<li>Read the results: <strong>Arithmetic progression<\/strong> lists all \\(n\\) terms, and <strong>Sum of first n terms<\/strong> gives their total.<\/li>\n<\/ol>\n<h2>What Is an Arithmetic Progression?<\/h2>\n<p>An arithmetic progression (AP), also called an arithmetic sequence, is a list of numbers in which each term differs from the previous one by the same constant \\(d\\), the common difference. The sequence 2, 5, 8, 11, 14, \u2026 is arithmetic because every step adds 3; the sequence 2, 4, 8, 16, \u2026 is not, because the ratio (not the difference) is constant \u2014 that one is a <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/geometric-mean\/\">geometric<\/a> pattern.<\/p>\n<p>Two values fully define an AP: the first term \\(a_1\\) and the common difference \\(d\\). From them you can reconstruct any term with \\(a_n = a_1 + (n-1)d\\), which means you can find the 100th term without writing out the 99 before it. The sign of \\(d\\) decides the behavior: a positive difference makes the sequence grow, a negative one makes it shrink, and \\(d = 0\\) makes every term identical.<\/p>\n<p>The sum \\(S_n\\) answers questions like \u201chow much in total after n deposits or payments?\u201d Thanks to the identity \\(S_n = \\frac{n}{2}(a_1 + a_n)\\), the sum is simply the average of the first and last term multiplied by the count \u2014 the same idea behind an <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/average-calculator\/\">average<\/a> calculation. For example, with terms 5 through 15 the average of the endpoints is 10, and \\(6 \\times 10 = 60\\).<\/p>\n<p>Arithmetic progressions appear in many practical settings: equal monthly savings or installments, straight-line depreciation, seating rows that grow by a fixed number of seats, and ladder-style discounts or pay raises. Whenever a quantity changes by the same absolute amount each step, an AP describes it.<\/p>\n<h2>Tips, Assumptions and Limitations<\/h2>\n<ul>\n<li><strong>Inputs can be decimals or negative numbers.<\/strong> The default step is 1, but you can type values like \u22123 or 0.5; a fractional difference still produces a valid AP.<\/li>\n<li><strong>Large n produces long output.<\/strong> The calculator lists every term up to n, so very large values of n will return a very long list; the sum stays exact regardless.<\/li>\n<li><strong>Indexing starts at 1.<\/strong> Here \\(a_1\\) is the first term and \\(a_n\\) is the term in position n, the standard convention in school mathematics.<\/li>\n<li><strong>Verify a sequence you already have:<\/strong> subtract consecutive terms; if the differences are all equal to \\(d\\), the sequence is arithmetic. If instead the ratios are constant, you need geometric tools rather than this calculator.<\/li>\n<li>For related sums over number patterns, see the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/sum-consecutive-squares\/\">sum of consecutive squares<\/a> or <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/floyds-triangle-generator\/\">Floyd's triangle<\/a> calculators.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Enter the number of terms, the first term, and the common difference to generate the full arithmetic progression and the sum of its first n terms instantly.<\/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-61448","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\/61448","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=61448"}],"version-history":[{"count":5,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61448\/revisions"}],"predecessor-version":[{"id":66205,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61448\/revisions\/66205"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61448"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61448"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61448"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}