The Fibonacci Sequence

The Fibonacci Sequence MATH 100 Survey of Mathematical Ideas J. Robert Buchanan Department of Mathematics Summer 2018

The Fibonacci Sequence In 1202 Leonardo of Pisa (a.k.a Fibonacci) wrote a problem in a mathematics book: A man put a pair of rabbits in a cage. During the first month the rabbits produced no offspring, but each month thereafter produced one new pair of rabbits. If each new pair thus produced reproduces in the same manner, how many pairs of rabbits will there be at the end of one year?

The Fibonacci Sequence January February March April May June July August September October November December

The Fibonacci Sequence January 1 February March April May June July August September October November December

The Fibonacci Sequence January 1 February 1 March April May June July August September October November December

The Fibonacci Sequence January 1 February 1 March 2 April May June July August September October November December

The Fibonacci Sequence January 1 February 1 March 2 April 3 May June July August September October November December

The Fibonacci Sequence January 1 February 1 March 2 April 3 May 5 June July August September October November December

The Fibonacci Sequence January 1 February 1 March 2 April 3 May 5 June 8 July August September October November December

The Fibonacci Sequence January 1 February 1 March 2 April 3 May 5 June 8 July 13 August 21 September 34 October 55 November 89 December 144

Recursion Formula Let F n represent the Fibonacci number in the nth position in the sequence, then F 1 = 1 F 2 = 1 F n = F n 2 + F n 1 for n 3. The last equation is known as a recursion formula and defines new elements in the sequence in terms of elements that appeared earlier.

Recursion Formula Let F n represent the Fibonacci number in the nth position in the sequence, then F 1 = 1 F 2 = 1 F n = F n 2 + F n 1 for n 3. The last equation is known as a recursion formula and defines new elements in the sequence in terms of elements that appeared earlier. The Fibonacci sequence has many interesting phenomena associated with it.

Using the Recursion Formula In our earlier list we saw F 11 = 89 and F 12 = 144, so F 13 = F 11 + F 12 = 89 + 144 = 233.

Using the Recursion Formula In our earlier list we saw F 11 = 89 and F 12 = 144, so F 13 = F 11 + F 12 = 89 + 144 = 233. Use your i>clicker to enter 1. F 20

Using the Recursion Formula In our earlier list we saw F 11 = 89 and F 12 = 144, so F 13 = F 11 + F 12 = 89 + 144 = 233. Use your i>clicker to enter 1. F 20 2. F 22

Using the Recursion Formula In our earlier list we saw F 11 = 89 and F 12 = 144, so F 13 = F 11 + F 12 = 89 + 144 = 233. Use your i>clicker to enter 1. F 20 2. F 22 3. F 25

Using the Recursion Formula In our earlier list we saw F 11 = 89 and F 12 = 144, so F 13 = F 11 + F 12 = 89 + 144 = 233. Use your i>clicker to enter 1. F 20 2. F 22 3. F 25 4. F 30

Fibonacci Behavior 1. Choose any Fibonacci number after the first one. Square your choice. 2. Multiply the Fibonacci numbers immediately before and after your choice. 3. Subtract the smaller number from the larger. What is your result?

Fibonacci Behavior 1. Choose any Fibonacci number after the first one. Square your choice. 2. Multiply the Fibonacci numbers immediately before and after your choice. 3. Subtract the smaller number from the larger. What is your result? F 12 = 144 F12 2 = 144 2 = 20736 F 13 F 11 = (89)(233) = 20737 20737 20736 = 1

Observation (1 of 2) Find the sum of the squares of the first n Fibonacci numbers and examine the pattern.

Observation (1 of 2) Find the sum of the squares of the first n Fibonacci numbers and examine the pattern. F 2 1 = 1 2 F 2 1 + F 2 2 = 1 2 + 1 2 F 2 1 + F 2 2 + F 2 3 = 1 2 + 1 2 + 2 2 F 2 1 + F 2 2 + F 2 3 + F 2 4 = 1 2 + 1 2 + 2 2 + 3 2 F 2 1 + F 2 2 + F 2 3 + F 2 4 + F 2 5 = 1 2 + 1 2 + 2 2 + 3 2 + 5 2

Observation (2 of 2) 1 2 = 1 1 2 + 1 2 = 2 1 2 + 1 2 + 2 2 = 6 1 2 + 1 2 + 2 2 + 3 2 = 15 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40.

Observation (2 of 2) 1 2 = 1 = 1 1 = F 1 F 2 1 2 + 1 2 = 2 1 2 + 1 2 + 2 2 = 6 1 2 + 1 2 + 2 2 + 3 2 = 15 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40.

Observation (2 of 2) 1 2 = 1 = 1 1 = F 1 F 2 1 2 + 1 2 = 2 = 1 2 = F 2 F 3 1 2 + 1 2 + 2 2 = 6 1 2 + 1 2 + 2 2 + 3 2 = 15 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40.

Observation (2 of 2) 1 2 = 1 = 1 1 = F 1 F 2 1 2 + 1 2 = 2 = 1 2 = F 2 F 3 1 2 + 1 2 + 2 2 = 6 = 2 3 = F 3 F 4 1 2 + 1 2 + 2 2 + 3 2 = 15 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40.

Observation (2 of 2) 1 2 = 1 = 1 1 = F 1 F 2 1 2 + 1 2 = 2 = 1 2 = F 2 F 3 1 2 + 1 2 + 2 2 = 6 = 2 3 = F 3 F 4 1 2 + 1 2 + 2 2 + 3 2 = 15 = 3 5 = F 4 F 5 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40.

Observation (2 of 2) 1 2 = 1 = 1 1 = F 1 F 2 1 2 + 1 2 = 2 = 1 2 = F 2 F 3 1 2 + 1 2 + 2 2 = 6 = 2 3 = F 3 F 4 1 2 + 1 2 + 2 2 + 3 2 = 15 = 3 5 = F 4 F 5 1 2 + 1 2 + 2 2 + 3 2 + 5 2 = 40 = 5 8 = F 5 F 6.

Fibonacci Sequence in Fractions (1 of 2) Since F 11 = 89 consider the fraction 1/89.

Fibonacci Sequence in Fractions (1 of 2) Since F 11 = 89 consider the fraction 1/89. 1 89 = 0.011235955056179775281...

Fibonacci Sequence in Fractions (2 of 2) 0.01 0.001 0.0002 0.00003 0.000005 0.0000008 0.00000013 0.000000021 0.0000000034 0.00000000055 0.000000000089 +. 0.011235955056179775281... = 1/89

Fibonacci Sequence in Pine Cones Consider the clockwise and counter-clockwise spirals of scales starting from the center on a pinecone. How many of each are there?

Two-Dimensional Pine Cone

Clockwise Spirals

Counter-clockwise Spirals

Honey Bees Male honeybees have only one parent (female). Female honeybees have two parents (female/male). The number of bees in each generation follows the Fibonacci sequence. Beginning with the 2nd generation, the number of female bees follows the Fibonacci sequence. Beginning with the 3rd generation, the number of male bees follows the Fibonacci sequence.

The Golden Ratio Consider the quotients of successive Fibonacci numbers. 1 1 = 1 2 1 = 2 3 2 = 1.5 5 3 = 1.6 8 5 = 1.6 13 8 = 1.625 21 13 = 1.615384 34 21 = 1.619047 55 34 1.617647059 89 55 = 1.618

The Golden Ratio Consider the quotients of successive Fibonacci numbers. 1 1 = 1 2 1 = 2 3 2 = 1.5 5 3 = 1.6 8 5 = 1.6 The quotients approach the number 13 8 = 1.625 21 13 = 1.615384 34 21 = 1.619047 55 34 1.617647059 89 55 = 1.618 1 + 5 2 1.6180339887498948482... which is known as the golden ratio.

Proof F n+1 F n = F n + F n 1 F n φ = 1 + 1 φ φ 2 φ 1 = 0 = 1 + F n 1 F n φ = 1 ± 1 4(1)( 1) 2 φ = 1 + 5 2

Golden Rectangle (1 of 3) A golden rectangle is one that can be divided into a square and another smaller rectangle the same shape as the original rectangle. W L L+W L L L

Golden Rectangle (2 of 3) Let the edge of the square be length L and let the height of the smaller rectangle be W, then L W = L + W L L W = L L + W L φ = 1 + 1 φ φ = 1 + 5. 2

Golden Rectangle (3 of 3) Artists, architects, and designers frequently use the golden rectangle.

Spiral