Fibonacci Sequences Susan Leggett, Zuzana Zvarova, Sara Campbell Fundamentals of Mathematics Mentor: Professor Foote

What Are Fibonacci Sequences? A series of numbers in which each number is the sum of the two preceding numbers, where by definition the first two numbers are 0 and 1 The sequence of Fibonacci numbers is defined by the recurrence relation: Fn= Fn-1 + Fn-2 Though found in many cultures, the sequences were named after Leonard of Pisa, also known as Fibonacci, after he published a book introducing the sequences to the western world 0,1,1,2,3,5,8,13,21,34,55,…

Applications Euclid’s Algorithm Hilbert’s Tenth Problem Used in pseudorandom number generators Computer programming Music Conversion factor Branching of trees and arrangement of fruit/flowers Bee ancestry code The Da Vinci Code Architecture

Fibonacci Identities Come from Combinatorial arguments F(n) can be interpreted as the number of sequences of 1s and 2s that have a sum of n-1 F(0) = 0 so that no sum will add to a negative value (empty sum will add to 0) Summands matters ( 1+2 and 2+1 are different)

Popular Identities of Fibonacci Sequences The nth Fibonacci number is the sum of the previous two Fibonacci numbers Fn=Fn-1+ Fn-2 The sum of the first n Fibonacci numbers is equal to the n+2nd Fibonacci number minus 1 Σfi=Fn+2-1 The sum of the first n-1 Fibonacci numbers, Fj, such that j is odd, is the (2n)th Fibonacci number. The sum of the first n Fibonacci numbers, Fj, such that j is even, is the (2n+1)th Fibonacci number minus 1 ΣF2i=F2n+1-1 ΣiFi= nFn+2- Fn+3+2 The sum of the squares of the first n Fibonacci numbers is the product of the nth and (n+1)th Fibonacci numbers. ΣFi2=FnFn+1

5th Identity Proof by Induction Inductive Hypothesis: Pn=  F2 = FnFn+1 Base Case: F0 = F1 = 1 P0 : 12 = 1 x 1 = 1 is true Assuming the inductive hypothesis for n = k Pk :  F2 = FkFk+1 We are trying to prove: Pk+1 :  F2 = Fk+1F(k+1)+1 = Fk+1Fk+2 (F0)2 + (F1)2 + … + (Fk)2 = FkFk+1 (F1)2 + … + (Fk)2 + (Fk+1)2 = FkFk+1 + (Fk+1)2  F2 = (Fk + Fk+1) Fk+1 Which gives us Pk+1 :  F2 = Fk+1 Fk+2 Hence by this proof by induction, for all n ≥ 0 we see that Pn is true

5th Identity Geometric Argument Fibonacci Rectangles Compute the area of the rectangles The n-th rectangle is composed of n squares with side lengths F1, F2, … Fn which is Pn=  F2 The n-th rectangle has side lengths Fn and Fn+1 which is Pn+1 =  F2 = Fn+2 Fn+1 Setting these expressions equal provides another proof 2 3 1 1 8 5

Divisibility Property Every kth number of the sequence is a multiple of 𝐹 𝑘 for example every 3rd number of the Fibonacci sequence is even Thus the Fibonacci sequence is an example of a divisibility sequence Satisfies the strong divisibility sequence gcd⁡(𝐹 𝑚 , 𝐹 𝑛 )= 𝐹 gcd⁡(𝑚,𝑛)

Right Triangles Starting with 5, every second Fibonacci number is the length of the hypotenuse of a right triangle with integer sides Every second Fibonacci number is the largest number of a Pythagorean triple 5x5 4x4 3x3

The Golden Ratio Consists of two quantities, a and b, such that a>b and φ ≈ 1.61803398874989 is the golden ratio, an irrational mathematical constant This constant is formally represented as The Golden Ratio appears in nature, such as leaf patterns, and math- especially geometry and Fibonacci numbers We have shown the Fibonacci Sequence as a linear recursion formula: Fn=Fn-1+ Fn-2 The closed form for the nth Fibonacci number is related to the Golden Ratio as follows: This closed-form expression is known as Binet’s formula

Golden Ratio and Fibonacci Numbers Proof is by induction, given and Fn=Fn-1+ Fn-2 Want to prove Binet’s Formula for all n Assuming that is true Show that Fk+1=Fk+ Fk-1 is true Proof by Induction is long, but our knowledge of induction is sufficient to understand it: http://fabulousfibonacci.com/portal/index.php?option=com_content&view=article&id=22&Itemid=22

Limit of Consecutive Fibonacci Numbers 8/5 = 1.6 , 13/8 = 1.625 , 21/13 = 1.615 … Johannes Kepler showed that these ratios converge to the Golden Ratio The proof involves substitution with Binet’s formula

Fibonacci Spiral Created by connecting opposite corners of Fibonacci squares of circular arcs The Fibonacci spiral and Fibonacci numbers occur in many aspects of nature, from seashells to flower petal arrangements, tree branching patterns, and reproduction in certain species

References Professor Foote http://en.wikipedia.org/wiki/Golden_ratio#Golden_ratio_conjugate http://en.wikipedia.org/wiki/Fibonacci_number http://fabulousfibonacci.com/portal/index.php?option=com_content&view=article&id=22&Itemid=22 http://www.fq.math.ca/Scanned/3-3/harris.pdf

Questions?

Homework Problem Calculate the first ten numbers in the Fibonacci Sequence. Do you see a pattern? (Show all work). Important Formula: Fn= Fn-1 + Fn-2