{"id":342,"date":"2018-09-14T20:08:56","date_gmt":"2018-09-14T20:08:56","guid":{"rendered":"http:\/\/www.sofadeve.com\/?p=342"},"modified":"2018-06-04T09:03:49","modified_gmt":"2018-06-04T09:03:49","slug":"patterns-in-the-fibonacci-numbers","status":"publish","type":"post","link":"https:\/\/sofadeve.com\/?p=342","title":{"rendered":"Patterns in the Fibonacci Numbers"},"content":{"rendered":"<p>Here are some patterns people have already noticed in the final digits of the Fibonacci numbers:<br \/>\nLook at the final digit in each Fibonacci number &#8211; the units digit:<br \/>\n0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, &#8230;<br \/>\nIs there a pattern in the final digits?<br \/>\n0, 1, 1, 2, 3, 5, 8,  3,  1,  4,  5,  9,   4,   3,   7,   0,   7, &#8230;<br \/>\nYes!<br \/>\nIt takes a while before it is noticeable. In fact, the series is just 60 numbers long and then it repeats the same sequence again and again all the way through the Fibonacci series &#8211; for ever. We say the series of final digits repeats with a cycle length of 60.<br \/>\nSuppose we look at the final two digits in the Fibonacci numbers. Do they have a pattern?<br \/>\n0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, &#8230;<br \/>\nYes, there is a pattern here too. After Fib(300) the last two digits repeat the same sequence again and again. The cycle length is 300 this time.<br \/>\nSo what about the last three digits?<br \/>\nand the last four digits?<br \/>\nand so on??<br \/>\nFor the last three digits, the cycle length is 1,500<br \/>\nfor the last four digits,the cycle length is 15,000 and<br \/>\nfor the last five digits the cycle length is 150,000<br \/>\nand so on&#8230;<\/p>\n<p><a href=\"http:\/\/www.maths.surrey.ac.uk\/hosted-sites\/R.Knott\/Fibonacci\/fibmaths.html\" target=\"_blank\">>> Go to Source Article<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Here are some patterns people have already noticed in the final digits of the Fibonacci numbers: Look at the final digit in each Fibonacci number &#8211; the units digit: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, &#8230; Is there a pattern in the final digits? [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":815,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"chat","meta":{"_et_pb_use_builder":"","_et_pb_old_content":"","_et_gb_content_width":"","_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"jetpack_post_was_ever_published":false},"categories":[30,4,31],"tags":[],"class_list":["post-342","post","type-post","status-publish","format-chat","has-post-thumbnail","hentry","category-knowledgebase","category-mathematics","category-patterns","post_format-post-format-chat"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/sofadeve.com\/wp-content\/uploads\/2014\/01\/fibonacci.jpeg","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p5jq4Y-5w","_links":{"self":[{"href":"https:\/\/sofadeve.com\/index.php?rest_route=\/wp\/v2\/posts\/342","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sofadeve.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sofadeve.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sofadeve.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sofadeve.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=342"}],"version-history":[{"count":4,"href":"https:\/\/sofadeve.com\/index.php?rest_route=\/wp\/v2\/posts\/342\/revisions"}],"predecessor-version":[{"id":820,"href":"https:\/\/sofadeve.com\/index.php?rest_route=\/wp\/v2\/posts\/342\/revisions\/820"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/sofadeve.com\/index.php?rest_route=\/wp\/v2\/media\/815"}],"wp:attachment":[{"href":"https:\/\/sofadeve.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=342"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sofadeve.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=342"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sofadeve.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=342"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}