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13.2: Arithmetic and Geometric Sequences

13.2: Arithmetic and Geometric Sequences. Recursive Definitions. 13.2 Recursive Definitions. Objectives : Write a recursive definition for a sequence Compare and contrast recursive and explicit Vocabulary : Sequence, term, recursive. Can you figure out the pattern?.

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13.2: Arithmetic and Geometric Sequences

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  1. 13.2: Arithmetic and Geometric Sequences Recursive Definitions

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

  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. 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

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

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

  9. Recursive Definition VS Explicit definition

  10. Recursive Definition For example,

  11. 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?

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

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

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

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

  16. Homework Textbook pg 481: #1-17odd

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