{"id":61495,"date":"2021-02-02T16:47:03","date_gmt":"2021-02-02T14:47:03","guid":{"rendered":"https:\/\/wpcalc.com\/?p=61495"},"modified":"2026-09-30T20:31:07","modified_gmt":"2026-09-30T17:31:07","slug":"fibonacci-series","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/fibonacci-series\/","title":{"rendered":"Fibonacci Series Generator"},"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\">Fibonacci Series Calculator<\/h2><\/div><div class=\"calcdash-favorite group\">\n  <button type=\"button\" class=\"favorite\" id=\"favorite\" data-favorite-id=\"61495\" data-favorite-title=\"Fibonacci Series \" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/fibonacci-series\/\" data-favorite-excerpt=\"Generate the Fibonacci sequence up to any n-th term and instantly get the sum of all terms. Enter n, and the calculator lists F(0) through F(n) using the recurrence F(n) = F(n-1) + F(n-2).\" 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\/fibonacci-series\/\" method=\"POST\" class=\"calculator\" id=\"calculator-61495\" data-post=\"61495\"><fieldset class=\"fieldset-input is-1\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field\" id=\"input-1\"><label for=\"field-1\">n\u1d57\u02b0 term (n)<\/label><input type=\"text\" name=\"n_term_n\" id=\"field-1\" data-number inputmode=\"decimal\" autocomplete=\"off\" spellcheck=\"false\" placeholder=\"10\" min=\"1\" value=\"10\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"61495\">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>Fibonacci total (sum)<\/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>Fibonacci sequence<\/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=\"61495\" 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>Fibonacci Number 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\\[ F(n) = F(n - 1) + F(n - 2), \\quad \\text{with} \\quad F(0) = 0, \\; F(1) = 1 \\]\n<\/div>\r\n\t<\/figure>\r\n\t<\/p>\n<p><strong>Where:<\/strong><\/p>\n<ul>\n<li>\\(F(n)\\) \u2014 the \\(n\\)-th Fibonacci number<\/li>\n<li>\\(F(n-1)\\), \\(F(n-2)\\) \u2014 the two previous terms of the sequence<\/li>\n<\/ul>\n<p>This recurrence relation defines each term as the sum of the two preceding ones, starting from 0 and 1. The sequence begins \\(0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, \\ldots\\)<\/p>\n<p>A closed-form expression also exists, known as Binet's formula:<\/p>\n<p>\\[ F(n) = \\frac{\\varphi^n - \\psi^n}{\\sqrt{5}}, \\quad \\varphi = \\frac{1 + \\sqrt{5}}{2}, \\; \\psi = \\frac{1 - \\sqrt{5}}{2} \\]<\/p>\n<p>Here \\(\\varphi \\approx 1.618\\) is the golden ratio, which the ratio of consecutive Fibonacci numbers approaches as \\(n\\) grows.<\/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>Fibonacci Series \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p><strong>Example:<\/strong> generate the Fibonacci series up to the 10th term, with \\(n = 10\\).<\/p>\n<p>\\[ F(0) = 0, \\; F(1) = 1 \\]<\/p>\n<p>\\[ F(2) = 1 + 0 = 1, \\quad F(3) = 1 + 1 = 2, \\quad F(4) = 2 + 1 = 3 \\]<\/p>\n<p>\\[ F(5) = 3 + 2 = 5, \\quad F(6) = 5 + 3 = 8, \\quad F(7) = 8 + 5 = 13 \\]<\/p>\n<p>\\[ F(8) = 13 + 8 = 21, \\quad F(9) = 21 + 13 = 34, \\quad F(10) = 34 + 21 = 55 \\]<\/p>\n<p><strong>Result:<\/strong> the sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and the total (sum) is 143.<\/p>\n<p>\n\r\n<\/section>\n<h2>How to Use the Fibonacci Series Calculator<\/h2>\n<ol>\n<li>Enter the desired term count in the <strong>\"n\u1d57\u02b0 term (n)\"<\/strong> field (for example, 10).<\/li>\n<li>The calculator generates the full sequence from \\(F(0)\\) up to \\(F(n)\\).<\/li>\n<li>Read the <strong>\"Fibonacci sequence\"<\/strong> result to see every term listed in order.<\/li>\n<li>Read the <strong>\"Fibonacci total (sum)\"<\/strong> result to get the sum of all generated terms.<\/li>\n<\/ol>\n<h2>What Is the Fibonacci Series?<\/h2>\n<p>The Fibonacci series is one of the most famous number sequences in mathematics. Each term is the sum of the two terms before it: starting from 0 and 1, you get 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The sequence is named after the Italian mathematician Leonardo of Pisa, known as Fibonacci, who introduced it to European mathematics in 1202 through his book <em>Liber Abaci<\/em>, where it modeled the growth of a rabbit population.<\/p>\n<p>The series has a remarkable property: as \\(n\\) increases, the ratio \\(F(n+1)\/F(n)\\) converges to the golden ratio \\(\\varphi \\approx 1.618\\). This connection links Fibonacci numbers to geometry, art, and architecture. In nature, Fibonacci numbers appear in the spirals of sunflowers and pinecones, the arrangement of leaves on a stem (phyllotaxis), and the branching patterns of trees.<\/p>\n<p>Beyond mathematics, the sequence is widely used in computer science. It illustrates recursion, dynamic programming, and memoization, and it appears in algorithms such as the Fibonacci search technique and the Fibonacci heap data structure. The sum of the first terms also has a neat closed form: the sum \\(F(0) + F(1) + \\cdots + F(n)\\) always equals \\(F(n+2) - 1\\). You can verify this with the example above: \\(F(12) - 1 = 144 - 1 = 143\\).<\/p>\n<p>If you are studying number sequences, you may also want to compare the Fibonacci series with a <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/geometric-progression\/\">Geometric Progression<\/a>, where each term is multiplied by a constant ratio instead of added.<\/p>\n<h2>Notes and Limitations<\/h2>\n<ul>\n<li>The calculator counts terms from \\(F(0) = 0\\), so entering \\(n = 10\\) produces \\(n + 1 = 11\\) numbers: \\(F(0)\\) through \\(F(10)\\).<\/li>\n<li>Values of \\(n\\) below 1 are invalid; the minimum allowed input is 1.<\/li>\n<li>Fibonacci numbers grow exponentially. For very large \\(n\\), the terms and their sum become huge, and the results are displayed as rounded JavaScript numbers, losing precision beyond \\(2^{53}\\) (around \\(n = 78\\)).<\/li>\n<li>Some sources start the sequence at 1, 1 instead of 0, 1. Both conventions describe the same underlying sequence; this calculator uses the standard indexing with \\(F(0) = 0\\).<\/li>\n<li>The sum result uses the identity \\(F(0) + F(1) + \\cdots + F(n) = F(n+2) - 1\\), which is a quick way to check the calculator's output by hand.<\/li>\n<\/ul>\n<h2>Quick Reference Table<\/h2>\n<table>\n<thead>\n<tr>\n<th>n<\/th>\n<th>F(n)<\/th>\n<th>Sum F(0)\u2026F(n)<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>1<\/td>\n<td>1<\/td>\n<td>1<\/td>\n<\/tr>\n<tr>\n<td>5<\/td>\n<td>5<\/td>\n<td>12<\/td>\n<\/tr>\n<tr>\n<td>10<\/td>\n<td>55<\/td>\n<td>143<\/td>\n<\/tr>\n<tr>\n<td>15<\/td>\n<td>610<\/td>\n<td>1,596<\/td>\n<\/tr>\n<tr>\n<td>20<\/td>\n<td>6,765<\/td>\n<td>17,710<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>For related tools, try the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/binomial-coefficient\/\">Binomial Coefficient calculator<\/a> or solve equations with the <a href=\"https:\/\/wpcalc.com\/en\/mathematics\/quadratic-equation-2\/\">Quadratic Equation calculator<\/a>.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Generate the Fibonacci sequence up to any n-th term and instantly get the sum of all terms. Enter n, and the calculator lists F(0) through F(n) using the recurrence F(n) = F(n-1) + F(n-2).<\/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-61495","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\/61495","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=61495"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61495\/revisions"}],"predecessor-version":[{"id":66237,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/61495\/revisions\/66237"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=61495"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=61495"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=61495"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}