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Section 13.2: Recursive Definitions

Section 13.2: Recursive Definitions. Pre-Calculus. Learning Targets. Write a recursive definition for a sequence Compare and contrast recursive definitions and explicit definitions. Can you figure out the pattern?. 1, 1, 2, 3, 5, 8, 13, 21, … What is the next number?.

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Section 13.2: Recursive Definitions

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  1. Section 13.2: Recursive Definitions Pre-Calculus

  2. Learning Targets • Write a recursive definition for a sequence • Compare and contrast recursive definitions and explicit 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. Recursive Definition • For example,

  7. Recursive Definition VS Explicit definition

  8. 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?

  9. Example 2:WRite a recursive definition • For the sequence: 1, 2, 6, 24, 120, 720, …

  10. Example 3:WRite a recursive definition • For the sequence: 3, 6, 12, 24, …

  11. Example 4:WRite a recursive definition • For the sequence: 4, 7, 13, 25, …

  12. Example 5:WRite a recursive definition • for the sequence: 10, 12, 15, 19, …

  13. Example 6:WRite a recursive definition • For the sequence: 1, 1, 2, 3, 5, 8, 13, 21, …

  14. 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

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

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

  17. Homework • Textbook pg 481: #1-3, 12, 13-17odd

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