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

2. 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,…

3. 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

4. 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)

5. 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

6. 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

7. 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

8. 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)

9. 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

10. The Golden Ratio
Consists of two quantities, a and b, such that a>b and
φ ≈ 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

11. 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:

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

13. 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

14. References Professor Foote

15. Questions?

16. 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