Fibonacci Sequences

Susan Leggett, Zuzana Zvarova, Sara

CampbellFundamentals 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

1. The nth Fibonacci number is the sum of the previous two Fibonacci numbers

Fn=Fn-1+ Fn-2

2. The sum of the first n Fibonacci numbers is equal to the n+2nd Fibonacci number minus 1

Σfi=Fn+2-1

3. 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

4. ΣiFi= nFn+2- Fn+3+2

5. 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

8

5

32

1 1

Divisibility Property

• Every kth number of the sequence is a

multiple of 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

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:

a ba

a

b

1 5

2

Fn n (1 )n

5This 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_conten

t&view=article&id=22&Itemid=22

•

Fn n (1 )n

5

1 5

2

Fk k (1 )k

5

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

nlim

Fn1Fn

nlim

Fn1Fn

nlim

n1 n1

n n

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=c

om_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