Chapter 9: Recursive Definitions, Fibonacci Numbers, and ...



Chapter 9: Recursive Definitions, Fibonacci Numbers, and the Golden Ratio

Recursive Sequence-defined by giving the value of tn (the term you’re for which you’re looking) in terms of the previous term, tn-1.

Examples:

1. t1=3 *These 2 lines are the “recursive definition”

tn=2tn-1+1

Find the next 3 terms in this sequence.

2. Find the recursive definition for the following sequence.

9, 13, 17, 21,…

9.1-Fibonacci Numbers

Each term in the sequence is the sum of the 2 previous terms.

If our first 2 terms are F1=1 and F2=1 and we use the information given above, find the next 8 terms of the Fibonacci Sequence.

What is the recursive definition?

Finding Fibonacci Numbers

Find: 1. F10

2. F10 + 2

3. F10+2

4. 3F4

9.2-Ф and x2=x+1

1. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

a. Find the ratio between each pair of Fibonacci numbers.

b. What does this ratio appear to be approaching as you go through the sequence?

2. Using the quadratic formula, solve for x in x2=x+1.

3. In x2=x+1 we found that x= Ф, so substitute.

Ф 2= Ф +1

a. Ф 3= *Hint: Multiply both sides of Ф 2= Ф +1

by Ф and simplify.

b. Ф 4=

c. Ф 5=

Do we see a pattern?

9.3-Similarity and Gnomons

When two figures are “similar,”

1. What is true about their side lengths?

2. What is true about their perimeters?

3. What is true about their areas?

These 2 figures are similar. Solve for x.

These 2 figures are similar. Solve for x.

[pic]

If the small figure has a perimeter of 30 units, what is the perimeter of the larger figure?

If the small figure has an area of 50 square units, what is the area of the larger figure?

Gnomon- define it yourself!!

[pic]

Find the value of x that will make “G” a Gnomon to our original figure “A.”

-----------------------

7

20

x

60

5

x+2

20

80

A

G

9

9

3

x

3

x

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