The Fibonacci Numbers A sequence of numbers (actually, positive integers) that occur many times in nature, art, music, geometry, mathematics, economics, and more.... The Fibonacci Numbers The numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Each Fibonacci number is the sum of the previous two Fibonacci numbers! Definition th The n Fibonacci number is written as Fn. F3 = 2, F5 = 5, F10 = 55 We have a recursive formula to find each Fibonacci number: Fn = Fn−1 + Fn−2 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . What makes these numbers so special? They occur many times in nature, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . What makes these numbers so special? They occur many times in nature, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . What makes these numbers so special? They occur many times in nature, art, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . What makes these numbers so special? They occur many times in nature, art, architecture, . 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? white Calla Lily 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? euphorbia 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? trillium 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? columbine 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? bloodroot 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? black-eyed susan 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? shasta daisy 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? field daisies 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? back of passion flower 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. How many petals are on each of these flowers? front of passion flower 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Let’s look at these Fibonacci numbers in nature. There are exceptions.... fuschia 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Looking more closely at the blossom in the center of the flower: What do we see? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Looking more closely at the blossom in the center of the flower: The stamen form spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Looking more closely at the blossom in the center of the flower: How many are there? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Looking more closely at the blossom in the center of the flower: How many are there? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Looking more closely at the blossom in the center of the flower: Not just in daisies. In Bellis perennis: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Looking more closely at the blossom in the center of the flower: Not just in daisies. In Bellis perennis: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . These numbers occur in other plants as well: The seed-bearing leaves of a simple pinecone: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . These numbers occur in other plants as well: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . These numbers occur in other plants as well: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . These numbers occur in other plants as well: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . These numbers occur in other plants as well: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Another pinecone: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Another pinecone: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . Another pinecone: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like cauliflower: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like cauliflower: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like cauliflower: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like romanesque: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like romanesque: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like romanesque: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like pineapples: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like pineapples: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like pineapples: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like pineapples: Find and count the number of spirals: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, . In fruits and vegetables, like pineapples: Find and count the number of spirals: The Fibonacci Numbers Let’s examine some interesting properties of these numbers. The numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Each Fibonacci number is the sum of the previous two Fibonacci numbers! th Let n any positive integer. If Fn is what we use to describe the n Fibonacci number, then Fn = Fn−1 + Fn−2 The Fibonacci Numbers For example, let’s look at the sum of the first several of these numbers. What is F1 + F2 + F3 + F4 ··· + Fn = ??? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Let’s look at the first few: F1 + F2 = 1 + 1 = 2 F1 + F2 + F3 = 1 + 1 + 2 = 4 F1 + F2 + F3 + F4 = 1 + 1 + 2 + 3 = 7 F1 + F2 + F3 + F4 + F5 = 1 + 1 + 2 + 3 + 5 = 12 F1 + F2 + F3 + F4 + F5 + F6 = 1 + 1 + 2 + 3 + 5 + 8 = 20 See a pattern? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... It looks like maybe the sum of the first few Fibonacci numbers is one less another Fibonacci number! But which Fibonacci number? F1 + F2 + F3 + F4 = 1 + 1 + 2 + 3 = 7 = F6 − 1 The sum of the first 4 Fibonacci numbers is one less the 6th Fibonacci number! F1 + F2 + F3 + F4 + F5 + F6 = 1 + 1 + 2 + 3 + 5 + 8 = 20 = F8 − 1 The sum of the first 6 Fibonacci numbers is one less the 8th Fibonacci number! 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... It looks like the sum of the first n Fibonacci numbers is one less the (n + 2)nd Fibonacci number. That is, F1 + F2 + F3 + F4 + ··· + Fn = Fn+2 − 1 Let’s check the formula, for n = 7: F1+F2+F3+F4+F5+F6+F7 = 1+1+2+3+5+8+13 = 33 = F9−1 It’s got hope to be true! Let’s try to prove it! 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377,... Conjecture For any positive integer n, the Fibonacci numbers satisfy: F1 + F2 + ··· + Fn = Fn+2 − 1 Let’s prove this (and then we’ll call it a Theorem.) Trying to prove: F1 + F2 + ··· Fn = Fn+2 − 1 We know F3 = F1 + F2.