{"id":61492,"date":"2021-02-02T16:33:08","date_gmt":"2021-02-02T14:33:08","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61492"},"modified":"2026-09-30T20:34:43","modified_gmt":"2026-09-30T17:34:43","slug":"geometric-progression","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/geometric-progression\/","title":{"rendered":"Geometric 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\">Find the n-th Term<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61492\" data-favorite-title=\"Geometric Progression \" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/geometric-progression\/\" data-favorite-excerpt=\"Find the n-th term of a geometric progression instantly. Enter the first term, the common ratio, and the term number, and the calculator returns the value of a\u2099 using a\u2099 = a\u2081 \u00b7 r\u207f\u207b\u00b9.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-algebra\">\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\/geometric-progression\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61492\" data-post=\"61492\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">First term (a\u2081)<\/label><input type=\"number\" name=\"first_term_a\" id=\"field-1\" placeholder=\"2\" step=\"1\" value=\"2\"\/><\/div><div class=\"field\" id=\"input-2\"><label for=\"field-2\">Common ratio (r)<\/label><input type=\"number\" name=\"common_ratio_r\" id=\"field-2\" placeholder=\"3\" step=\"1\" value=\"3\"\/><\/div><div class=\"field\" id=\"input-3\"><label for=\"field-3\">Term number (n)<\/label><input type=\"text\" name=\"term_number_n\" id=\"field-3\" 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=\"61492\">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>n\u1d57\u02b0 term (a\u2099)<\/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=\"61492\" 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\" 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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>Geometric 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\\[ a_n = a_1 \\cdot r^{\\,n-1} \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(a_n\\) \u2014 the n-th term of the progression<\/li>\n<li>\\(a_1\\) \u2014 the first term<\/li>\n<li>\\(r\\) \u2014 the common ratio (the constant each term is multiplied by)<\/li>\n<li>\\(n\\) \u2014 the term number (a positive integer)<\/li>\n<\/ul>\n<p>This formula gives the value of any term in a geometric progression directly, without computing all the terms before it. Each term equals the first term multiplied by the common ratio raised to the power \\(n-1\\).<\/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>Geometric Progression \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> a geometric progression starts at \\(a_1 = 2\\) and each term is 3 times the previous one (\\(r = 3\\)). Find the 5th term.<\/p>\n<p>\\[ a_5 = 2 \\cdot 3^{5-1} = 2 \\cdot 3^4 = 2 \\cdot 81 = 162 \\]<\/p>\n<p><strong>Result:<\/strong> \\(a_5 = 162\\). The full sequence is 2, 6, 18, 54, 162, \u2026<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Geometric Progression Calculator<\/h2>\n<ol>\n<li>Enter the <strong>First term (a\u2081)<\/strong> \u2014 the value the sequence starts with, for example 2.<\/li>\n<li>Enter the <strong>Common ratio (r)<\/strong> \u2014 the constant multiplier between consecutive terms, for example 3.<\/li>\n<li>Enter the <strong>Term number (n)<\/strong> \u2014 which term you want to find; it must be a positive integer (1 or greater).<\/li>\n<li>Read the result: the <strong>n\u1d57\u02b0 term (a\u2099)<\/strong> is calculated automatically as \\(a_1 \\cdot r^{n-1}\\).<\/li>\n<\/ol>\n<p>The result is rounded to three decimal places, which is enough for most classroom and practical problems. If you enter a term number smaller than 1, the calculator reports \u201cInvalid n\u201d.<\/p>\n<h2>What Is a Geometric Progression?<\/h2>\n<p>A geometric progression (also called a geometric sequence) is a sequence of numbers in which each term after the first is obtained by multiplying the previous term by a fixed non-zero constant called the <em>common ratio<\/em> \\(r\\). For example, with \\(a_1 = 2\\) and \\(r = 3\\), the sequence is 2, 6, 18, 54, 162, \u2026 The same ratio 3 appears between every pair of consecutive terms: \\(6\/2 = 18\/6 = 54\/18 = 3\\).<\/p>\n<p>The key property of a geometric progression is exponential behavior. If \\(|r| &gt; 1\\), the terms grow (or grow in magnitude) very quickly; if \\(0 &lt; |r| &lt; 1\\), the terms shrink toward zero; if \\(r = 1\\), every term equals the first term; and if \\(r\\) is negative, the terms alternate in sign. This makes geometric progressions the natural model for anything that changes by a fixed percentage or factor per step \u2014 compound interest, population growth, radioactive decay, or repeatedly halving a quantity.<\/p>\n<p>The formula \\(a_n = a_1 \\cdot r^{n-1}\\) lets you jump straight to any term without listing the earlier ones. That is exactly what this calculator does: finding the 50th term of a progression takes the same effort as finding the 5th. If you also need the sum of the first \\(n\\) terms, it is given separately by \\(S_n = a_1 \\frac{r^n - 1}{r - 1}\\) (for \\(r \\neq 1\\)), which you can evaluate with the same inputs.<\/p>\n<h2>Interpreting the Result<\/h2>\n<p>The result \\(a_n\\) is simply the value of the term at position \\(n\\) in the sequence. Keep in mind how the inputs interact:<\/p>\n<ul>\n<li>A larger \\(|r|\\) makes the terms grow or shrink faster; even small changes in \\(r\\) have large effects on distant terms.<\/li>\n<li>A negative \\(r\\) produces alternating signs, e.g. \\(a_1 = 2\\), \\(r = -3\\) gives 2, \u22126, 18, \u221254, \u2026<\/li>\n<li>Fractional ratios such as \\(r = 0.5\\) model decay, e.g. 8, 4, 2, 1, 0.5, \u2026<\/li>\n<\/ul>\n<p>Always sanity-check the result: for \\(n = 1\\) the answer must equal \\(a_1\\), and for \\(n = 2\\) it must equal \\(a_1 \\cdot r\\).<\/p>\n<h2>Assumptions, Limitations and Tips<\/h2>\n<ul>\n<li><strong>n must be a positive integer.<\/strong> The calculator works with term positions 1, 2, 3, \u2026 and rejects values below 1.<\/li>\n<li><strong>r can be any real number<\/strong> \u2014 positive, negative, fractional, or 1. A ratio of exactly 1 makes every term identical.<\/li>\n<li><strong>Rounding:<\/strong> the answer is rounded to three decimal places. With very large \\(n\\) or large \\(|r|\\), results are shown in decimal notation and may be enormous.<\/li>\n<li><strong>This tool finds a single term.<\/strong> To sum the whole progression, apply the sum formula \\(S_n = a_1 \\frac{r^n - 1}{r - 1}\\) with the same inputs.<\/li>\n<li><strong>Check for typos in r.<\/strong> Because the ratio is raised to a power, entering 0.3 instead of 3 changes the answer dramatically for large \\(n\\).<\/li>\n<\/ul>\n<p>Geometric progressions often appear alongside other algebra topics. If you are working through series and sequences, you may also find our <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/fibonacci-series\/\">Fibonacci series calculator<\/a> useful, or our tools for the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/proportion-calculator\/\">proportion calculator<\/a> and <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/binomial-coefficient\/\">binomial coefficient<\/a> when studying related counting and ratio problems.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Find the n-th term of a geometric progression instantly. Enter the first term, the common ratio, and the term number, and the calculator returns the value of a\u2099 using a\u2099 = a\u2081 \u00b7 r\u207f\u207b\u00b9.<\/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":[11],"class_list":["post-61492","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-algebra"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61492","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=61492"}],"version-history":[{"count":4,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61492\/revisions"}],"predecessor-version":[{"id":66248,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61492\/revisions\/66248"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61492"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61492"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61492"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}