Fibonacci Sequence and Related Fibonacci Sequence and Related NumbersNumbers

• The famous Fibonacci sequence is the result of The famous Fibonacci sequence is the result of a question posed by Leonardo de Fibonacci, a a question posed by Leonardo de Fibonacci, a mathematician during the Middle Ages.mathematician during the Middle Ages.

If you begin with one pair of rabbits on the first If you begin with one pair of rabbits on the first day of the year, how many pairs of rabbits will day of the year, how many pairs of rabbits will you have on the first day of the next year?you have on the first day of the next year?

It is assumed that each pair of rabbits produces a It is assumed that each pair of rabbits produces a new pair every month and each new pair begins to new pair every month and each new pair begins to produce two months after birth. produce two months after birth.

The Fibonacci ProblemThe Fibonacci Problem

How Many Pairs of Rabbits Are How Many Pairs of Rabbits Are Created by One Pair in One YearCreated by One Pair in One Year

A certain man had one pair of rabbits A certain man had one pair of rabbits together in a certain enclosed place, and together in a certain enclosed place, and one wishes to know how many are created one wishes to know how many are created from the pair in one year when it is the from the pair in one year when it is the nature of them in a single month to bear nature of them in a single month to bear another pair, and in the second month another pair, and in the second month those born to bear also. those born to bear also.

Fibonacci cont’lFibonacci cont’l He died in the 1240's and there is now a He died in the 1240's and there is now a

statue commemorating him located at the statue commemorating him located at the Leaning Tower end of the cemetery next Leaning Tower end of the cemetery next to the Cathedral in Pisa.to the Cathedral in Pisa.

Fig. 2-69, p. 113

Fig. 2-70, p. 114

Fibonacci Sequence, cont’dFibonacci Sequence, cont’d• The solution to this question is shown in the table The solution to this question is shown in the table

below.below.• The sequence that appears three times in the The sequence that appears three times in the

table, 1, 1, 2, 3, 5, 8, 13, 21, … is called the table, 1, 1, 2, 3, 5, 8, 13, 21, … is called the Fibonacci sequenceFibonacci sequence. .

Fibonacci Sequence, cont’dFibonacci Sequence, cont’d

• The The Fibonacci sequenceFibonacci sequence is the sequence is the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …of numbers 1, 1, 2, 3, 5, 8, 13, 21, …

• The Fibonacci sequence is found in many The Fibonacci sequence is found in many places in nature.places in nature.

• Any number in the sequence is called a Any number in the sequence is called a Fibonacci number.Fibonacci number.

• The sequence is usually written The sequence is usually written F F11, F, F22, F, F33, …, F, …, Fnn, …, …

RecursionRecursion• RecursionRecursion, in a sequence, indicates that , in a sequence, indicates that

each number in the sequence is found each number in the sequence is found using previous numbers in the sequence.using previous numbers in the sequence.

• Some sequences, such as the Fibonacci Some sequences, such as the Fibonacci sequence, are generated by a sequence, are generated by a recursion recursion rulerule along with along with starting valuesstarting values for the first for the first two, or more, numbers in the sequence.two, or more, numbers in the sequence.

Fibonacci Numbers In NatureFibonacci Numbers In Nature• The Fibonacci numbers are found many The Fibonacci numbers are found many

places in the natural world, including:places in the natural world, including:• The number of flower petals.The number of flower petals.• The branching behavior of plants.The branching behavior of plants.• The growth patterns of sunflowers and The growth patterns of sunflowers and

pinecones, ……pinecones, ……• It is believed that the spiral nature of plant It is believed that the spiral nature of plant

growth accounts for this phenomenon.growth accounts for this phenomenon.

Fibonacci Numbers In Nature, cont’dFibonacci Numbers In Nature, cont’d• The number of petals on a flower are The number of petals on a flower are

often Fibonacci numbers.often Fibonacci numbers.

Fibonacci Numbers In Nature, cont’dFibonacci Numbers In Nature, cont’d• Plants grow in a spiral pattern. The ratio of the Plants grow in a spiral pattern. The ratio of the

number of spirals to the number of branches is number of spirals to the number of branches is called the called the phyllotactic ratiophyllotactic ratio..

• The numbers in the phyllotactic ratio are usually The numbers in the phyllotactic ratio are usually Fibonacci numbers.Fibonacci numbers.

Fibonacci Numbers In Nature, cont’dFibonacci Numbers In Nature, cont’d

• Mature sunflowers have one set of spirals Mature sunflowers have one set of spirals going clockwise and another set going going clockwise and another set going counterclockwise.counterclockwise.

• The numbers of spirals in each set are The numbers of spirals in each set are usually a pair of adjacent Fibonacci usually a pair of adjacent Fibonacci numbers.numbers.• The most common number of spirals is 34 The most common number of spirals is 34

and 55. and 55.