13.2: Arithmetic and Geometric Sequences

Recursive Definitions

Objectives:1. Write a recursive definition for a sequence

2. Compare and contrast recursive and explicit

Vocabulary:Sequence, term, recursive

13.2 Recursive Definitions

Can you figure out the pattern?

1, 1, 2, 3, 5, 8, 13, 21, …

What is the next number?

1, 1, 2, 3, 5, 8, 13, 21, …

To find the next number in the sequence, we had to know the previous numbers.

This idea is called a recursive definition

Recursive Definition

A definition of a sequence that consists ofAn initial condition to indicate the start of the sequenceA formula that relates how any term in the sequence is

related to the preceding term

Recursion is one of the central ideas of computer scienceTo solve a problem recursively Break it down into smaller subproblems each having

the same form as the original problem; When the process is repeated many times, the last of

the subproblems are small and easy to solve; The solutions of the subproblems can be woven

together to form a solution to the original problem.

Example: The tower of Hanoi (P. 484 #31)

Fibonacci Sequence One of the most FAMOUS sequences of all

time! It generates the golden ratio which is believed to

be the “ratio of god” and is very evident in the world around us.

The golden ratio is produced by dividing any number in the sequence by its preceding term

Fibonacci Sequence It is used in

Graphic DesignArt MusicArchitecturePhotographyPersonal BeautyStock Market Analysis

Fibonacci Sequence

http://www.youtube.com/watch?v=0tAZe6pP-FM

Recursive Definition VS Explicit definition

Explicit Definition Recursive Definition

1. Plug in exact nth term2. Can find any term in the

sequence just by knowing its location in the sequence

1. Dependent on the previous term(s)

2. Reflect the pattern of the sequence very clearly

Recursive Definition

For example,

Example 1: Finding the next terms in the sequence Given find the next 3 terms in the sequence

Is it possible to write an explicit formula for this sequence? If so, what is it?

Example 2:Write a recursive definition For the sequence: 1, 2, 6, 24, 120, 720, …

Explicit?

Example 3:Write a recursive definition For the sequence: 3, 6, 12, 24, …

Explicit?

Example 4:Write a recursive definition For the sequence: 4, 7, 13, 25, …

Explicit?

Example 5:Write a recursive definition for the sequence: 10, 12, 15, 19, …

Explicit?

Homework

Textbook pg 481: #5-21odd, 32