{"id":224,"date":"2019-06-26T23:05:23","date_gmt":"2019-06-26T23:05:23","guid":{"rendered":"http:\/\/funfacts.104.42.120.246.xip.io\/?page_id=224"},"modified":"2019-10-18T23:10:17","modified_gmt":"2019-10-18T23:10:17","slug":"fibonacci-number-formula","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/funfacts\/fibonacci-number-formula\/","title":{"rendered":"Fibonacci Number Formula"},"content":{"rendered":"\n<p>The Fibonacci numbers are generated by setting F<sub>0<\/sub> = 0, F<sub>1 <\/sub>= 1, and then using the recursive formula<br>F<sub>n<\/sub>\u00a0= F<sub>n-1<\/sub>\u00a0+ F<sub>n-2<\/sub><br>to get the rest. Thus the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, &#8230; This sequence of\u00a0Fibonacci numbers arises all over mathematics and also in nature.<\/p>\n\n\n\n<p>However, if I wanted the 100th term of this sequence, it would take lots of intermediate calculations with the recursive formula to get a result. Is there an easier way?<\/p>\n\n\n\n<p>Yes, there is an exact formula for the n-th term! It is:<br>a<sub>n<\/sub>\u00a0= [Phi<sup>n<\/sup>\u00a0&#8211; (phi)<sup>n<\/sup>] \/ Sqrt[5].<br>where Phi = (1 + Sqrt[5]) \/ 2 is the so-called\u00a0golden mean, and <br>phi = (1 &#8211; Sqrt[5]) \/ 2 is an associated golden number, also equal to (-1 \/ Phi). This formula is attributed to Binet in 1843, though known by Euler before him.<\/p>\n\n\n\n<p><strong>The&nbsp;Math&nbsp;Behind&nbsp;the&nbsp;Fact:<\/strong><br>The formula can be proved by induction. It can also be proved using the&nbsp;eigenvalues&nbsp;of a 2&#215;2-matrix that encodes the recurrence. You can learn more about recurrence formulas in a fun course called discrete mathematics.<\/p>\n\n\n\n<p><strong>How to Cite this Page:<\/strong>&nbsp;<br>Su, Francis E., et al. &#8220;Fibonacci Number Formula.&#8221;&nbsp;<em>Math Fun Facts<\/em>. &lt;http:\/\/www.math.hmc.edu\/funfacts&gt;.<\/p>\n\n\n\n<p><strong>Fun Fact suggested by:   <\/strong><br>Arthur Benjamin <\/p>\n","protected":false},"excerpt":{"rendered":"<p>The Fibonacci numbers are generated by setting F0 = 0, F1 = 1, and then using the recursive formulaFn\u00a0= Fn-1\u00a0+&#46;&#46;&#46;<\/p>\n","protected":false},"author":7,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[106,9,3,76,10,105],"class_list":["post-224","page","type-page","status-publish","hentry","tag-binets-formula","tag-combinatorics","tag-easy","tag-fibonacci","tag-numtheory","tag-proof-by-induction"],"jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/224","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/users\/7"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/comments?post=224"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/224\/revisions"}],"predecessor-version":[{"id":1329,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/pages\/224\/revisions\/1329"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/media?parent=224"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/funfacts\/wp-json\/wp\/v2\/tags?post=224"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}