mth 105 sec 2.3 notes.notebook - linn–benton …cf.linnbenton.edu/mathsci/math/rogerss/upload/mth...

MTH 105 Sec 2.3 notes.notebook April 12, 2011

Section 2.3Fibonacci Numbers and the

Golden Mean• Goals• Study the Fibonacci Sequence• Recursive sequences• Fibonacci number occurrences in nature• Geometric recursion• The golden ratio


2.3 Initial Problem• This expression is called a continued fraction.

• How can you find the exact decimal equivalent of this number?• The solution will be given at the end of the section.


Sequences• A sequence is an ordered collection of numbers.• A sequence can be written in the form a1, a2, a3, …, an, …• The symbol a1 represents the first number in the sequence.• The symbol an represents the nth number in the sequence.


Question:

Given the sequence: 1, 3, 5, 7, 9, 11, 13, 15, … , find the values of the numbers A1, A3, and A9.

a. A1 = 1, A3 = 5, A9 = 15b. A1 = 1, A3 = 3, A9 = 17c. A1 = 1, A3 = 5, A9 = 17d. A1 = 1, A3 = 5, A9 = 16


Fibonacci Sequence• The famous Fibonacci sequence is the result of a question posed by Leonardo de Fibonacci, a mathematician during the Middle Ages.• If you begin with one pair of rabbits on the first day of the year, how many pairs of rabbits will you have on the first day of the next year?• It is assumed that each pair of rabbits produces a new pair every month and each new pair begins to produce two months after birth.


Fibonacci Sequence, cont’d• The solution to this question is shown in the table below.• The sequence that appears three times in the table, 1, 1, 2, 3, 5, 8, 13, 21, … is called the Fibonacci sequence.


Fibonacci Sequence, cont’d• The Fibonacci sequence is the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …• The Fibonacci sequence is found many places in nature.• Any number in the sequence is called a Fibonacci number.• The sequence is usually written f1, f2, f3, …, fn, …


Recursion• Recursion, in a sequence, indicates that each number in the sequence is found using previous numbers in the sequence.• Some sequences, such as the Fibonacci sequence, are generated by a recursion rule along with starting values for the first two, or more, numbers in the sequence.


Write a recursion rule to generate the Fibonacci Sequence:


Question:

A recursive sequence uses the rule An =4An1 – An2, with starting values of A1 = 2, A2 =7.

What is the fourth term in the sequence?

a. A4 = 45 c. A4 = 67b. A4 = 26 d. A4 = 30


Fibonacci Sequence, cont’d• For the Fibonacci sequence, the starting values are f1 = 1 and f2 = 1. • The recursion rule for the Fibonacci sequence is:

• Example: Find the third number in the sequence using the formula.• Let n = 3.


Example 1• Suppose a tree starts from one shoot that grows for two months and then sprouts a second branch. If each established branch begins to spout a new branch after one month’s growth, and if every new branch begins to sprout its own first new branch after two month’s growth, how many branches does the tree have at the end of the year?


Example 1, cont’d• Solution: The number of branches each month in the first year is given in the table and drawn in the figure below.


Fibonacci Numbers In Nature• The Fibonacci numbers are found many places in the natural world, including:• The number of flower petals.• The branching behavior of plants.• The growth patterns of sunflowers and pinecones.• It is believed that the spiral nature of plant growth accounts for this phenomenon.


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


Fibonacci Numbers In Nature, cont’d• Plants grow in a spiral pattern. The ratio of the number of spirals to the number of branches is called the phyllotactic ratio.• The numbers in the phyllotactic ratio are usually Fibonacci numbers.


Fibonacci Numbers In Nature, cont’d

• Example: The branch at right has a phyllotactic ratio of 3/8. • Both 3 and 8 are Fibonacci numbers.


Fibonacci Numbers In Nature, cont’d

• Mature sunflowers have one set of spirals going clockwise and another set going counterclockwise.• The numbers of spirals in each set are usually a pair of adjacent Fibonacci numbers.• The most common number of spirals is 34 and 55.


Geometric Recursion

• In addition to being used to generate a sequence, the recursion process can also be used to create shapes.• The process of building a figure stepbystep by repeating a rule is called geometric recursion.


Example 2• Beginning with a 1by1 square, form a sequence of rectangles by adding a square to the bottom, then to the right, then to the bottom, then to the right, and so on.• Draw the resulting rectangles.• What are the dimensions of the rectangles?


Draw the Squares and Rectangles as previously described:


Example 2, cont’d• Solution: • The first seven rectangles in the sequence are shown below.


Example 2, cont’d• Solution cont’d: •

Notice that the dimensions of each rectangle are consecutive


The Golden Ratio

• Consider the ratios of pairs of consecutive Fibonacci numbers. • Some of the ratios are calculated in the table shown on the following slide.


The Golden Ratio, cont’d


The Golden Ratio, cont’d• The ratios of pairs of consecutive Fibonacci numbers are also represented in the graph below.• The ratios approach the dashed line which represents a number around 1.618.


The Golden Ratio, cont’d• The irrational number, approximately 1.618, is called the golden ratio.• Other names for the golden ratio include the golden section, the golden mean, and the divine proportion.• The golden ratio is represented by the Greek letter φ, which is pronounced “fe” or “fi”.


The Golden Ratio, cont’d• The golden ratio has an exact value of

• The golden ratio has been used in mathematics, art, and architecture for more than 2000 years.


Golden Rectangles

• A golden rectangle has a ratio of the longer side to the shorter side that is the golden ratio.• Golden rectangles are used in architecture, art, and packaging.


Golden Rectangles, cont’d• The rectangle enclosing the diagram of the Parthenon is an example of a golden rectangle.


Creating a Golden Rectangle

• Start with a square, WXYZ, that measures one unit on each side.• Label the midpoint of side WX as point M.


Creating a Golden Rectangle, cont’d

• Draw an arc centered at M with radius MY.• Label the point P as shown.


Creating a Golden Rectangle, cont’d

• Draw a line perpendicular to WP.• Extend ZY to meet this line, labeling point Q as shown. The completed rectangle is shown.


2.3 Initial Problem Solution• How can you find the exact decimal equivalent of this number?


Initial Problem Solution, cont’d• We can find the value of the continued fraction by using a recursion rule that generates a sequence of fractions.• The first term is

• The recursion rule is


Initial Problem Solution, cont’d

• We find:• The first term is

• The second term is


Initial Problem Solution, cont’d

• The third term is

• The fourth term is


Initial Problem Solution, cont’d• The fractions in this sequence are

2, 3/2, 5/3, 8/5, … • This is recognized to be the same as the ratios of consecutive pairs of Fibonacci numbers.• The numbers in this sequence of fractions get closer and closer to φ.


Example:See Exercise #2 on page 125

Sequence: 1, 8, 27, 64, 125, 216, 343, 512

Note interesting problems and topics pp. 130-133!

mth 105 sec 2.3 notes.notebook - linn–benton …cf.linnbenton.edu/mathsci/math/rogerss/upload/mth...

Documents