{"id":25302,"date":"2023-02-23T03:52:24","date_gmt":"2023-02-23T03:52:24","guid":{"rendered":"https:\/\/www.splashlearn.com\/math-vocabulary\/?page_id=25302"},"modified":"2024-02-05T16:05:49","modified_gmt":"2024-02-05T16:05:49","slug":"fibonacci-sequence","status":"publish","type":"post","link":"https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence","title":{"rendered":"Fibonacci Sequence &#8211; Definition, Formula, Examples"},"content":{"rendered":"<div class=\"aioseo-breadcrumbs\"><span class=\"aioseo-breadcrumb\">\n\tMath-Vocabulary\n<\/span><\/div>\n\n<div class=\"ub_table-of-contents\" data-showtext=\"show\" data-hidetext=\"hide\" data-scrolltype=\"auto\" id=\"ub_table-of-contents-06dabfce-d1d9-49f2-b93f-7d9352aa8738\" data-initiallyhideonmobile=\"false\"\n                    data-initiallyshow=\"true\"><div class=\"ub_table-of-contents-extra-container\"><div class=\"ub_table-of-contents-container ub_table-of-contents-1-column \"><ul><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence#0-what-is-the-fibonacci-sequence>What Is the Fibonacci Sequence?<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence#2-fibonacci-sequence-formula>Fibonacci Sequence Formula<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence#4-fibonacci-sequence-properties>Fibonacci Sequence Properties<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence#8-solved-examples-on-fibonacci-sequence>Solved Examples on Fibonacci Sequence<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence#9-practice-problems-on-fibonacci-sequence>Practice Problems on Fibonacci Sequence<\/a><\/li><li><a href=https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence#10-frequently-asked-questions-on-fibonacci-sequence>Frequently Asked Questions on Fibonacci Sequence<\/a><\/li><\/ul><\/div><\/div><\/div>\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"0-what-is-the-fibonacci-sequence\">What Is the Fibonacci Sequence?<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In mathematics, we define the sequence as an ordered list of numbers that follow a particular pattern. The numbers that are present in the sequence are also known as the terms.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">One such interesting sequence is given below. What do you observe? Can you identify the pattern?<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"206\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-1.png\" alt=\"Identifying the pattern of the Fibonacci sequence\" class=\"wp-image-25344\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-1.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-1-300x100.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">This sequence is known as the \u2018Fibonacci sequence\u2019 in math. If you look closely at the numbers, you can see that each number is the sum of two previous numbers.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Let\u2019s learn the definition, formula, properties, and some interesting facts.<\/p>\n\n\n\n<h3 class=\"wp-block-heading eplus-wrapper\" id=\"1-fibonacci-sequence-definition\">Fibonacci Sequence Definition<\/h3>\n\n\n\n<p class=\"eplus-wrapper\">The <strong>Fibonacci sequence is an infinite sequence that starts with 0 and 1 and continues in such a way that each number is the sum of the previous two numbers<\/strong>.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The numbers in the Fibonacci sequence are also known as Fibonacci numbers. The following image shows the examples of fibonacci numbers and explains their pattern.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"561\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-2.png\" alt=\"Generating Fibonacci Numbers\" class=\"wp-image-25345\" title=\"Generating Fibonacci Numbers\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-2.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-2-300x271.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">The Fibonacci Sequence is given as:<\/p>\n\n\n\n<p class=\"eplus-wrapper\">0, 1, 1, 2, 3, 5, 8, 13, 21, \u2026<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Here, we obtain the third term \u201c1\u201d by adding the first and second term. $(0 + 1 = 1)$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Similarly, we obtain \u201c2\u201d by adding the second and third term. $(1 + 1 = 2)$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">So, the term after 21 will be the sum of 13 and 21, i.e., $13 + 21 = 34$. Therefore, the next term in the sequence is 34.&nbsp;<\/p>\n\n\n\n<div id=\"recommended-games-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Games<\/h4><div class=\"recommended-games-container-slides\"><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/choose-the-next-number-in-the-counting-sequence\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_count_forward_w5_pt.png\" alt=\"Choose the Next Number in the Counting Sequence Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Choose the Next Number in the Counting Sequence Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/click-on-the-next-number-in-the-counting-sequence\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_sequence_1_w10_pt.png\" alt=\"Click on the Next Number in the Counting Sequence Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Click on the Next Number in the Counting Sequence Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/complete-the-counting-sequence\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_missing_numbers_w5_pt.png\" alt=\"Complete the Counting Sequence Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Complete the Counting Sequence Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/complete-the-number-sequence\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_skip5_complete_pattern_1_pt.png\" alt=\"Complete the Number Sequence Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Complete the Number Sequence Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/complete-the-sequence\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_complete_seq_2_pt.png\" alt=\"Complete the Sequence Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Complete the Sequence Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/complete-the-skip-counting-sequence\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_skip5_complete_pattern_2_pt.png\" alt=\"Complete the Skip Counting Sequence Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Complete the Skip Counting Sequence Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/count-back-and-complete-the-sequence\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_comp_count_back_pt.png\" alt=\"Count back and Complete the Sequence Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Count back and Complete the Sequence Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/counting-sequence-from-1-to-5\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/count_stack_house_9_gm.png\" alt=\"Counting Sequence from 1 to 5 Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Counting Sequence from 1 to 5 Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/number-sequence-from-1-to-10\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/num_seq_stack_house_7_gm.png\" alt=\"Number Sequence from 1 to 10 Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Number Sequence from 1 to 10 Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><div class=\"game-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-games\/number-sequence-from-1-to-20\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"games_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t\t<div class=\"game-card-container-inner-block\">\r\n\t\t\t<img decoding=\"async\" class=\"game-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/curriculum_uploads\/images\/playables\/num_seq_stack_house_12_gm.png\" alt=\"Number Sequence from 1 to 20 Game\">\r\n\t\t<\/div>\r\n\t\r\n\t\t<div class=\"game-card-container-inner\">\r\n\t\t\t<div class=\"game-card-container-inner-name\">Number Sequence from 1 to 20 Game<\/div>\r\n\t\t\t<span class=\"game-card-container-inner-cta\">Play<\/span>\r\n\t\t<\/div>\r\n\t<\/a>\r\n<\/div><\/div><p class=\"recommended-games-container-desc\"><a href=\"https:\/\/www.splashlearn.com\/games\">More Games<\/a><\/p><button class=\"scroll-right-arrow\"><\/button><\/div><script type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-games-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".game-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"2-fibonacci-sequence-formula\">Fibonacci Sequence Formula<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The Fibonacci sequence of numbers, say \u201cFn\u201d where the suffix n denotes the order or rank of term, is defined by&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Initial term: $F_{0} = 0$&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">First term: $F_{1} = 1$&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">These two terms together are known as the kick-off part.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Formula for next terms: $F_{n} = F_{n \\;-\\; 1} + F_{n \\;-\\; 2}$&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">This is known as a recursive formula since we repeat the same rule to get the next terms.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$F_{2} = F_{1} + F_{0}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$F_{3} = F_{2} + F_{1}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$F_{4} = F_{3} + F_{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$F_{5} = F_{4} + F_{3&#8242;}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">and so on.<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Note:<\/strong> The sequence starts with 0 (not 1). So, the 4<sup>th<\/sup> term of the sequence will be $F_{3}$.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The kick-off part is $F_{0} = 0$ and $F_{1} = 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The recursive relation part is $F_{n} = F_{n\\;-\\;1} + F_{n\\;-\\;2}$<\/p>\n\n\n\n<div id=\"recommended-worksheets-container-id\" class=\"recommended-games-container\"><h4 class=\"recommended-games-container-headline\">Recommended Worksheets<\/h4><div class=\"recommended-games-container-slides\"><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/classify-sorted-and-unsorted-number-sequences\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/classify-sorted-and-unsorted-number-sequences.jpeg\" alt=\"Classify Sorted and Unsorted Number Sequences Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/classify-the-number-sequences\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/classify-the-number-sequences.jpeg\" alt=\"Classify the Number Sequences Worksheet\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/counting-sequence-within-10-with-christmas-cookies\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/counting-sequence-within-10-with-christmas-cookies.jpeg\" alt=\"Counting Sequence within 10 with Christmas Cookies\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/counting-sequence-within-10-with-clouds\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/counting-sequence-within-10-with-clouds.jpeg\" alt=\"Counting Sequence within 10 with Clouds\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/counting-sequence-within-10-with-dinosaurs\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/counting-sequence-within-10-with-dinosaurs.jpeg\" alt=\"Counting Sequence within 10 with Dinosaurs\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a href=\"https:\/\/www.splashlearn.com\/s\/math-worksheets\/counting-sequence-within-10-with-donuts\" data-vars-ga-category=\"splashlearn_vocab\" data-vars-ga-action=\"worksheets_recommendations\" data-vars-ga-label=\"post_widget\">\r\n\t    <div class=\"worksheet-card-container-inner-block\">\r\n\t        <img decoding=\"async\" class=\"worksheet-card-container-inner-img\" src=\"https:\/\/cdn.splashmath.com\/cms_assets\/s\/math-worksheets\/counting-sequence-within-10-with-donuts.jpeg\" alt=\"Counting Sequence within 10 with Donuts\">\r\n\t    <\/div>\r\n\t\t<div class=\"worksheet-card-container-inner\" >\r\n\t\t<\/div><\/a>\r\n<\/div><div class=\"worksheet-card-container-outer\">\r\n\t<a 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type=\"text\/javascript\">\n        document.addEventListener(\"DOMContentLoaded\", function() {\n            const container = document.querySelector(\"#recommended-worksheets-container-id\");\n            const slidesContainer = container.querySelector(\".recommended-games-container-slides\");\n            const cards = slidesContainer.querySelectorAll(\".worksheet-card-container-outer\");\n            const scrollRightArrow = container.querySelector(\".scroll-right-arrow\");\n\n            function adjustContainerStyles() {\n                const numCards = cards.length;\n\n                if (numCards === 1) {\n                    container.style.maxWidth = \"30%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 2) {\n                    container.style.maxWidth = \"50%\";\n                    container.style.textAlign = \"center\";\n                } else if (numCards === 3) {\n                    container.style.maxWidth = \"75%\";\n                    container.style.textAlign = \"center\";\n                } else {\n                    container.style.maxWidth = \"\";\n                    container.style.textAlign = \"\";\n                }\n            }\n\n            function checkScrollPosition() {\n                const maxScrollLeft = slidesContainer.scrollWidth - slidesContainer.clientWidth;\n                if ((slidesContainer.scrollLeft + 10) >= maxScrollLeft) {\n                    scrollRightArrow.style.display = \"none\"; \/\/ Hide the arrow if fully scrolled\n                } else {\n                    scrollRightArrow.style.display = \"block\"; \/\/ Show the arrow if not fully scrolled\n                }\n            }\n\n            scrollRightArrow.addEventListener(\"click\", function() {\n                const scrollAmount = 300; \/\/ Adjust based on the container's width and your needs\n                slidesContainer.scrollLeft += scrollAmount;\n                setTimeout(checkScrollPosition, 100); \/\/ Delay to allow scroll update\n            });\n\n            adjustContainerStyles();\n            checkScrollPosition();\n            slidesContainer.addEventListener(\"scroll\", checkScrollPosition);\n        });\n    <\/script><h2 class=\"wp-block-heading eplus-wrapper\" id=\"3-fibonacci-sequence-list\">Fibonacci Sequence List<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The list of fibonacci numbers (up to F<sub>20<\/sub>) is given below:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"613\" height=\"574\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-3.png\" alt=\"Fibonacci numbers\" class=\"wp-image-25346\" title=\"Fibonacci numbers\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-3.png 613w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-3-300x281.png 300w\" sizes=\"auto, (max-width: 613px) 100vw, 613px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"4-fibonacci-sequence-properties\">Fibonacci Sequence Properties<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The properties of the Fibonacci sequence are given as follows:<\/p>\n\n\n\n<ol class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Fibonacci numbers are related to the Golden ratio. In mathematics, two quantities are said to be in golden ratio if their ratio is equal to the ratio of their sum to the larger of the two quantities. It is commonly found in nature, and also in mathematics, design, geometry, etc.<br>The ratio of any two successive Fibonacci numbers is close to the Golden ratio, which is approximately equal to \u200b\u200b1.618034. Any Fibonacci number can be calculated using the golden ratio.<\/li>\n<\/ol>\n\n\n\n<ol start=\"2\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">&nbsp;The ratio of consecutive Fibonacci terms is given below:<\/li>\n<\/ol>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"619\" height=\"515\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-4.png\" alt=\"Ratio of consecutive Fibonacci numbers\" class=\"wp-image-25348\" title=\"Ratio of consecutive Fibonacci numbers\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-4.png 619w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-4-300x250.png 300w\" sizes=\"auto, (max-width: 619px) 100vw, 619px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">By multiplying the previous Fibonacci Number using the golden ratio, i.e., 1.618034, we get the approximated Fibonacci number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">For example, 8 is the sixth term in the sequence. To find the seventh term, we multiply 8 by the golden ratio.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$8 \\times 1.618034&#8230; = 12.94427\u2026 \\approx 13$.<\/p>\n\n\n\n<ol start=\"3\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The terms of the Fibonacci sequence below zero can also be calculated by:&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">$F_{-n} = (\\;-\\;1)^{n+1}\\; F_{n}$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">For example: $F_{-3} = (\\;-\\;1)^{3+1} \\times 2 = 2$.&nbsp;<\/p>\n\n\n\n<ol start=\"4\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">There is an interesting pattern in the sequence.&nbsp;<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">Every 3rd number in the sequence starting from 2 is a multiple of 2.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Examples: $F_{3} = 2,\\; F_{6} = 8$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Every 4th number in the sequence starting from 3 is a multiple of 3.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Examples: $F_{4} = 3,\\; F_{8} = 21$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Every 5th number starting from 5 is a multiple of 5; and so on.<\/p>\n\n\n\n<ol start=\"5\" class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">The sum of n terms of Fibonacci Sequence is given by:<\/li>\n<\/ol>\n\n\n\n<p class=\"eplus-wrapper\">$\\sum_{i=0}^{n}\\;F_{i} = F_{n+2}\\;-\\;F_{2} = F_{n+2}\\;-\\; 1$, where Fn is the n<sup>th<\/sup> Fibonacci number.&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">For example, the sum of the first 5 terms of sequence $= \\sum_{i=0}^{4}\\;F_{i} = F_{4+2}\\;-\\;F_{2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$= F_{6}\\;-\\;1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$= 8 \\;\u2013\\; 1$&nbsp;<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$= 7$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"5-fibonacci-spiral\">Fibonacci Spiral<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">Fibonacci spiral is a geometric pattern or a spiral formed with squares having sides representing the numbers in the Fibonacci sequence. The squares fit together due to the pattern in which Fibonacci numbers occur and thus form a spiral.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Connecting the corners of the squares, the spiral is drawn. The larger the numbers are in the Fibonacci sequence, the closet the ratio becomes to the golden ratio.&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"524\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-5.png\" alt=\"Fibonacci spiral\" class=\"wp-image-25349\" title=\"Fibonacci spiral\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-5.png 620w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-5-300x254.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/figure>\n\n\n\n<p class=\"eplus-wrapper\">In this Fibonacci spiral, every two consecutive terms represent the length and breadth of a rectangle. The following are the ratios of every two successive terms of the Fibonacci sequence.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">This spiral can also be observed in nature. Take a look!<\/p>\n\n\n\n<figure class=\"wp-block-image size-full eplus-wrapper\"><img loading=\"lazy\" decoding=\"async\" width=\"613\" height=\"445\" src=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-6.png\" alt=\"Fibonacci spirals in the nature\" class=\"wp-image-25350\" title=\"Fibonacci spirals in the nature\" srcset=\"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-6.png 613w, https:\/\/www.splashlearn.com\/math-vocabulary\/wp-content\/uploads\/2023\/02\/Fibonacci-Sequence-6-300x218.png 300w\" sizes=\"auto, (max-width: 613px) 100vw, 613px\" \/><\/figure>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"6-applications-of-the-fibonacci-sequence-in-real-life\">Applications of the Fibonacci Sequence in Real Life<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">The Fibonacci sequence has many applications due to its unique pattern and relation with the golden ratio.<\/p>\n\n\n\n<ul class=\"eplus-wrapper wp-block-list\">\n<li class=\"eplus-wrapper\">Computer algorithms, coding theories, security coding, and data structures use Fibonacci numbers.<br><\/li>\n\n\n\n<li class=\"eplus-wrapper\">Many things in nature follow the Fibonacci sequence. Examples: Branching in trees, arrangement of leaves, pine cone structure.<br><\/li>\n\n\n\n<li class=\"eplus-wrapper\">It is also used in numerous fields of science that includes high-energy physical science, quantum mechanics, cryptography, etc.<br><\/li>\n\n\n\n<li class=\"eplus-wrapper\">Finance, trading, and stock markets use the Fibonacci pattern for technical analysis.<\/li>\n<\/ul>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"7-conclusion\">Conclusion<\/h2>\n\n\n\n<p class=\"eplus-wrapper\">In this article, we learned about the Fibonacci sequence, which is a sequence in which a term is the sum of the terms before that. Let\u2019s solve a few examples for better understanding!<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"8-solved-examples-on-fibonacci-sequence\">Solved Examples on Fibonacci Sequence<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><strong>1. Find the 11th term of the Fibonacci series if the 9th and 10th terms are 34 and 55 respectively.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution: <\/strong>11th term $=$ 9<sup>th<\/sup> term + 10<sup>th<\/sup> term, i.e.,<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$F_{11} = F_{10} + F_{9} = 55 + 34 = 89$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>2. Find the Fibonacci number when <\/strong>$n = 7$<strong>, using the recursive relation.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution: <\/strong>The formula to calculate the Fibonacci number is: $F_{n} = F_{n\\;-\\;1} + F_{n\\;-\\;2}$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">Taking $F_{0} = 0$ and $F_{1} = 1$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$F_{2} = 0 + 1 = 1,\\; F_{3} = 1 + 1 = 2,\\; F_{4} = 2 + 1 = 3,\\; F_{5} = 3 + 2 = 5,\\; F_{6} = 5 +3 = 8$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$F_{7} = 8 + 5 = 13$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>3.<\/strong><strong> Find the sum of the first 6 Fibonacci terms.<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong> The formula to calculate the sum of the first 6 Fibonacci terms is:<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\sum_{i=0}^{n}\\;F_{i} = F_{n+2}\\;-\\;F_{2} = F_{n+2}\\;-\\;1$, where $F_{n}$ is the n<sup>th<\/sup> Fibonacci number<\/p>\n\n\n\n<p class=\"eplus-wrapper\">$\\sum_{i=0}^{5}\\;F_{i} = F_{5+2}\\;-\\;F_{2} = F_{7}\\;-\\;1 = 13\\;-\\;1 = 12$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>4. <\/strong><strong>Find the missing terms in the Fibonacci sequence:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>0, 1, 1, 2, 3, 5, 8, 13, 21, ?, ?, \u2026<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\">The next term after 21 will be $21 + 13 = 34$.<\/p>\n\n\n\n<p class=\"eplus-wrapper\">The next term after 34 is $34 + 21 = 55$<\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>5. Find the value of <\/strong><strong>14th<\/strong><strong><sup> <\/sup><\/strong><strong>and 15th terms in the Fibonacci sequence if the 12th<\/strong><strong><sup> <\/sup><\/strong><strong>and 13th terms are 144 and 233 respectively.&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"eplus-wrapper\"><strong>Solution:<\/strong> 14<sup>th<\/sup> term $=$ 13<sup>th<\/sup> term + 12<sup>th<\/sup> term $= 233 + 144 = 377$<\/p>\n\n\n\n<p class=\"eplus-wrapper\">15<sup>th<\/sup> term $=$ 14<sup>th<\/sup> term $+$ 13<sup>th<\/sup> term $= 377 + 233 = 610$<\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"9-practice-problems-on-fibonacci-sequence\">Practice Problems on Fibonacci Sequence<\/h2>\n\n\n\n<p class=\"eplus-wrapper\"><div class=\"spq_wrapper\"><h2 style=\"display:none;\">Fibonacci Sequence<\/h2><p style=\"display:none;\">Attend this quiz & Test your knowledge.<\/p><div class=\"spq_question_wrapper\" data-answer=\"1\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">1<\/span><h3 class=\"sqp_question_text\">The first term of the Fibonacci sequence is<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$1$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$0$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$-\\;1$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$2$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $0$<br\/>The first term in Fibonacci Sequence is $F_{0} = 0$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"0\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">2<\/span><h3 class=\"sqp_question_text\">If the $16^{th}$ term in the Fibonacci series is 610. Find the next term of the series.<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">987<\/div><div class=\"spq_answer_block\" data-value=\"1\">988<\/div><div class=\"spq_answer_block\" data-value=\"2\">990<\/div><div class=\"spq_answer_block\" data-value=\"3\">1000<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 987<br\/>$16^{th}$ term $= 610$. The next term, i.e., $17^{th}$ term $= 610 \\times 1.618 = 986.98 \\approx 987$<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">3<\/span><h3 class=\"sqp_question_text\">Which of the following is the approximate value of the golden ratio?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">1.6078<\/div><div class=\"spq_answer_block\" data-value=\"1\">1.612<\/div><div class=\"spq_answer_block\" data-value=\"2\">1.618034<\/div><div class=\"spq_answer_block\" data-value=\"3\">1.67803<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 1.618034<br\/>The approximate value of the golden rule is 1.618034.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"2\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">4<\/span><h3 class=\"sqp_question_text\">What is the value of $F_{-\\;4}$ if $F_{4} = 3$?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">$\u2013\\; 4$<\/div><div class=\"spq_answer_block\" data-value=\"1\">$\u2013\\; 2$<\/div><div class=\"spq_answer_block\" data-value=\"2\">$\u2013\\; 3$<\/div><div class=\"spq_answer_block\" data-value=\"3\">$3$<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: $\u2013\\; 3$<br\/>$F_{-\\;4} = (\\;-\\;1)^{4+1} 3 = \\;-\\;3$.<\/span><\/div><\/div><\/div><div class=\"spq_question_wrapper\" data-answer=\"3\"><span class=\"spq_question_header\"><span class=\"sqp_question_number\">5<\/span><h3 class=\"sqp_question_text\">What is the sum of the first 7 terms of the Fibonacci sequence?<\/h3><\/span><div class=\"spq_answer_block\" data-value=\"0\">27<\/div><div class=\"spq_answer_block\" data-value=\"1\">29<\/div><div class=\"spq_answer_block\" data-value=\"2\">33<\/div><div class=\"spq_answer_block\" data-value=\"3\">20<\/div><div class=\"sqp_question_hint\"><div class=\"sqp_question_hint__header\"><span class=\"spq_correct\">Correct<\/span><span class=\"spq_incorrect\">Incorrect<\/span><\/div><div class=\"sqp_question_hint__content\"><span>Correct answer is: 20<br\/>Sum of first 7 terms $= F_{6+2}\\;-\\;F_{2} = 8^{th}$ term $\\;\u2013\\; 1 = 21 \\;\u2013\\; 1 = 20$.<\/span><\/div><\/div><\/div><\/div>  <script type=\"application\/ld+json\">{\n        \"@context\": \"https:\/\/schema.org\/\", \n        \"@type\": \"Quiz\", \n        \"typicalAgeRange\": \"3-11\",\n        \"educationalLevel\":  \"beginner\",\n        \"assesses\" : \"Attend this quiz & Test your knowledge.\",\n        \"educationalAlignment\": [\n              {\n                \"@type\": \"AlignmentObject\",\n                \"alignmentType\": \"educationalSubject\",\n                \"targetName\": \"Math\"\n              }] ,\n        \"name\": \"Fibonacci Sequence\",        \n        \"about\": {\n                \"@type\": \"Thing\",\n                \"name\": \"Fibonacci Sequence\"\n        },  \n        \"hasPart\": [{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"The first term of the Fibonacci sequence is\",\n                    \"text\": \"The first term of the Fibonacci sequence is\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The first term in Fibonacci Sequence is $$F_{0} = 0$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$1$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The first term in Fibonacci Sequence is $$F_{0} = 0$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$-\\\\;1$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The first term in Fibonacci Sequence is $$F_{0} = 0$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$2$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The first term in Fibonacci Sequence is $$F_{0} = 0$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 1,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$0$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The first term in Fibonacci Sequence is $$F_{0} = 0$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The first term in Fibonacci Sequence is $$F_{0} = 0$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"If the $$16^{th}$$ term in the Fibonacci series is 610. Find the next term of the series.\",\n                    \"text\": \"If the $$16^{th}$$ term in the Fibonacci series is 610. Find the next term of the series.\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$16^{th}$$ term $$= 610$$. The next term, i.e., $$17^{th}$$ term $$= 610 \\\\times 1.618 = 986.98 \\\\approx 987$$\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"988\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$16^{th}$$ term $$= 610$$. The next term, i.e., $$17^{th}$$ term $$= 610 \\\\times 1.618 = 986.98 \\\\approx 987$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"990\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$16^{th}$$ term $$= 610$$. The next term, i.e., $$17^{th}$$ term $$= 610 \\\\times 1.618 = 986.98 \\\\approx 987$$\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1000\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$16^{th}$$ term $$= 610$$. The next term, i.e., $$17^{th}$$ term $$= 610 \\\\times 1.618 = 986.98 \\\\approx 987$$\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 0,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"987\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$16^{th}$$ term $$= 610$$. The next term, i.e., $$17^{th}$$ term $$= 610 \\\\times 1.618 = 986.98 \\\\approx 987$$\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$16^{th}$$ term $$= 610$$. The next term, i.e., $$17^{th}$$ term $$= 610 \\\\times 1.618 = 986.98 \\\\approx 987$$\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"Which of the following is the approximate value of the golden ratio?\",\n                    \"text\": \"Which of the following is the approximate value of the golden ratio?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"The approximate value of the golden rule is 1.618034.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1.6078\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The approximate value of the golden rule is 1.618034.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1.612\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The approximate value of the golden rule is 1.618034.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"1.67803\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"The approximate value of the golden rule is 1.618034.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"1.618034\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"The approximate value of the golden rule is 1.618034.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"The approximate value of the golden rule is 1.618034.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the value of $$F_{-\\\\;4}$$ if $$F_{4} = 3$$?\",\n                    \"text\": \"What is the value of $$F_{-\\\\;4}$$ if $$F_{4} = 3$$?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"$$F_{-\\\\;4} = (\\\\;-\\\\;1)^{4+1} 3 = \\\\;-\\\\;3$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\u2013\\\\; 4$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$F_{-\\\\;4} = (\\\\;-\\\\;1)^{4+1} 3 = \\\\;-\\\\;3$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$\u2013\\\\; 2$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$F_{-\\\\;4} = (\\\\;-\\\\;1)^{4+1} 3 = \\\\;-\\\\;3$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 3,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"$$3$$\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"$$F_{-\\\\;4} = (\\\\;-\\\\;1)^{4+1} 3 = \\\\;-\\\\;3$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 2,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"$$\u2013\\\\; 3$$\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"$$F_{-\\\\;4} = (\\\\;-\\\\;1)^{4+1} 3 = \\\\;-\\\\;3$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"$$F_{-\\\\;4} = (\\\\;-\\\\;1)^{4+1} 3 = \\\\;-\\\\;3$$.\"\n                      }\n                    } \n\n                    },{\n                    \"@type\": \"Question\",   \n                    \"eduQuestionType\": \"Multiple choice\",\n                    \"learningResourceType\": \"Practice problem\",\n                    \"name\": \"What is the sum of the first 7 terms of the Fibonacci sequence?\",\n                    \"text\": \"What is the sum of the first 7 terms of the Fibonacci sequence?\",\n                    \"comment\": {\n                      \"@type\": \"Comment\",\n                      \"text\": \"Sum of first 7 terms $$= F_{6+2}\\\\;-\\\\;F_{2} = 8^{th}$$ term $$\\\\;\u2013\\\\; 1 = 21 \\\\;\u2013\\\\; 1 = 20$$.\"\n                    },\n                    \"encodingFormat\": \"text\/html\",\n                    \"suggestedAnswer\": [ {\n                                \"@type\": \"Answer\",\n                                \"position\": 0,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"27\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Sum of first 7 terms $$= F_{6+2}\\\\;-\\\\;F_{2} = 8^{th}$$ term $$\\\\;\u2013\\\\; 1 = 21 \\\\;\u2013\\\\; 1 = 20$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 1,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"29\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Sum of first 7 terms $$= F_{6+2}\\\\;-\\\\;F_{2} = 8^{th}$$ term $$\\\\;\u2013\\\\; 1 = 21 \\\\;\u2013\\\\; 1 = 20$$.\"\n                                    }\n                                }, {\n                                \"@type\": \"Answer\",\n                                \"position\": 2,\n                                \"encodingFormat\": \"text\/html\",\n                                \"text\": \"33\",\n                                \"comment\": {\n                                    \"@type\": \"Comment\",\n                                    \"text\": \"Sum of first 7 terms $$= F_{6+2}\\\\;-\\\\;F_{2} = 8^{th}$$ term $$\\\\;\u2013\\\\; 1 = 21 \\\\;\u2013\\\\; 1 = 20$$.\"\n                                    }\n                                }],\n                    \"acceptedAnswer\": {\n                      \"@type\": \"Answer\",\n                      \"position\": 3,\n                      \"encodingFormat\": \"text\/html\",\n                      \"text\": \"20\",\n                      \"comment\": {\n                          \"@type\": \"Comment\",\n                          \"text\": \"Sum of first 7 terms $$= F_{6+2}\\\\;-\\\\;F_{2} = 8^{th}$$ term $$\\\\;\u2013\\\\; 1 = 21 \\\\;\u2013\\\\; 1 = 20$$.\"\n                        },\n                      \"answerExplanation\": {\n                        \"@type\": \"Comment\",\n                        \"text\": \"Sum of first 7 terms $$= F_{6+2}\\\\;-\\\\;F_{2} = 8^{th}$$ term $$\\\\;\u2013\\\\; 1 = 21 \\\\;\u2013\\\\; 1 = 20$$.\"\n                      }\n                    } \n\n                    }]}<\/script><\/p>\n\n\n\n<h2 class=\"wp-block-heading eplus-wrapper\" id=\"10-frequently-asked-questions-on-fibonacci-sequence\">Frequently Asked Questions on Fibonacci Sequence<\/h2>\n\n\n<div class=\"wp-block-ub-content-toggle\" id=\"ub-content-toggle-a452ce19-0c6a-477c-8000-7d6060652109\" data-mobilecollapse=\"true\" data-desktopcollapse=\"true\">\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-0-a452ce19-0c6a-477c-8000-7d6060652109\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a452ce19-0c6a-477c-8000-7d6060652109\"><strong>What are the examples of the Fibonacci series in nature?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-0-a452ce19-0c6a-477c-8000-7d6060652109\">\n\n<p class=\"eplus-wrapper\">The Fibonacci series can be spotted in the nature around us in different forms. It can be found in the spirals of the petals of certain flowers such as in the flower heads of sunflowers. We can also find them in the branches of trees.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-1-a452ce19-0c6a-477c-8000-7d6060652109\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a452ce19-0c6a-477c-8000-7d6060652109\"><strong>What is the 100<sup>th<\/sup> term of the Fibonacci series?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-1-a452ce19-0c6a-477c-8000-7d6060652109\">\n\n<p class=\"eplus-wrapper\">The 100<sup>th<\/sup> term in a Fibonacci series is 354,224,848,179,261,915,075. Using the recursion formula, the 100<sup>th<\/sup> term is the sum of the 98<sup>th<\/sup> and 99<sup>th<\/sup> terms.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-2-a452ce19-0c6a-477c-8000-7d6060652109\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a452ce19-0c6a-477c-8000-7d6060652109\"><strong>What is the importance of the Fibonacci series?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-2-a452ce19-0c6a-477c-8000-7d6060652109\">\n\n<p class=\"eplus-wrapper\">The Fibonacci series is important because it is related to the golden ratio and Pascal&#8217;s triangle. Except for the initial numbers, the numbers in the series have a pattern that each number $\\approx 1.618$ times its previous number. The value becomes closer to the golden ratio as the number of terms in the Fibonacci series increases.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-3-a452ce19-0c6a-477c-8000-7d6060652109\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a452ce19-0c6a-477c-8000-7d6060652109\"><strong>How is the Fibonacci sequence related to humans?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-3-a452ce19-0c6a-477c-8000-7d6060652109\">\n\n<p class=\"eplus-wrapper\">The number of bones of your finger (from knuckles to wrist) are based on the Fibonacci sequence. Facial proportions are also based on the golden ratio. Human eye finds any object featuring the golden ratio appealing and beautiful.<\/p>\n\n<\/div><\/div>\n\n<div class=\"wp-block-ub-content-toggle-accordion\">\n                <div class=\"wp-block-ub-content-toggle-accordion-title-wrap\" aria-expanded=\"false\" aria-controls=\"ub-content-toggle-panel-4-a452ce19-0c6a-477c-8000-7d6060652109\" tabindex=\"0\">\n                    <p class=\"wp-block-ub-content-toggle-accordion-title ub-content-toggle-title-a452ce19-0c6a-477c-8000-7d6060652109\"><strong>What fruits show a Fibonacci pattern?<\/strong><\/p><div class=\"wp-block-ub-content-toggle-accordion-toggle-wrap right\"><span class=\"wp-block-ub-content-toggle-accordion-state-indicator wp-block-ub-chevron-down\"><\/span>\n                    <\/div><\/div><div role=\"region\" class=\"wp-block-ub-content-toggle-accordion-content-wrap ub-hide\" id=\"ub-content-toggle-panel-4-a452ce19-0c6a-477c-8000-7d6060652109\">\n\n<p class=\"eplus-wrapper\">Fruits like the pineapple, banana, persimmon, apple and others exhibit patterns that follow the Fibonacci sequence.<\/p>\n\n<\/div><\/div>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>What Is the Fibonacci Sequence? In mathematics, we define the sequence as an ordered list of numbers that follow a particular pattern. The numbers that are present in the sequence are also known as the terms.&nbsp; One such interesting sequence is given below. What do you observe? Can you identify the pattern? This sequence is &#8230; <a title=\"Fibonacci Sequence &#8211; Definition, Formula, Examples\" class=\"read-more\" href=\"https:\/\/www.splashlearn.com\/math-vocabulary\/fibonacci-sequence\" aria-label=\"More on Fibonacci Sequence &#8211; Definition, Formula, Examples\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ub_ctt_via":"","footnotes":""},"categories":[1],"tags":[],"class_list":["post-25302","post","type-post","status-publish","format-standard","hentry","category-all"],"featured_image_src":null,"author_info":{"display_name":"Mithun Jhawar","author_link":"https:\/\/www.splashlearn.com\/math-vocabulary\/author\/mithun-jhawarsplashlearn-com\/"},"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/25302","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/comments?post=25302"}],"version-history":[{"count":12,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/25302\/revisions"}],"predecessor-version":[{"id":39951,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/posts\/25302\/revisions\/39951"}],"wp:attachment":[{"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/media?parent=25302"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/categories?post=25302"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.splashlearn.com\/math-vocabulary\/wp-json\/wp\/v2\/tags?post=25302"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}