1. 13.2: Arithmetic and Geometric Sequences Recursive Definitions

2. Objectives : 1. Write a recursive definition for a sequence 2. Compare and contrast recursive and explicit Vocabulary : Sequence, term, recursive 13.2 Recursive Definitions

3. Can you figure out the pattern? 1, 1, 2, 3, 5, 8, 13, 21, … What is the next number?

4. 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

5. Recursive Definition A definition of a sequence that consists of  An initial condition to indicate the start of the sequence  A formula that relates how any term in the sequence is related to the preceding term

6. Recursion is one of the central ideas of computer science To 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)

7. 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

8. Fibonacci Sequence It is used in  Graphic Design  Art  Music  Architecture  Photography  Personal Beauty  Stock Market Analysis

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

10. Recursive Definition VS Explicit definition Explicit DefinitionRecursive Definition 1.Plug in exact n th term 2.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

11. Recursive Definition

12. Example 1: Finding the next terms in the sequence

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

14. Example 3: Write a recursive definition For the sequence: 3, 6, 12, 24, … Explicit?

15. Example 4: Write a recursive definition For the sequence: 4, 7, 13, 25, … Explicit?

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

17. Homework Textbook pg 481: #5-21odd, 32