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? 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 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

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 Design Art Music Architecture Photography Personal Beauty Stock Market Analysis

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

Recursive Definition VS Explicit definition Plug in exact nth term Can find any term in the sequence just by knowing its location in the sequence Dependent on the previous term(s) 2. Reflect the pattern of the sequence very clearly

Recursive Definition For example, 𝑡 1 =3, 𝑎𝑛𝑑 𝑡 𝑛 =2 𝑡 𝑛−1 +1 𝑡 1 =3 𝑡 2 =2 𝑡 2−1 +1=2 𝑡 1 +1=2 3 +1=7 𝑡 3 =2 𝑡 3−1 +1=2 𝑡 2 +1=2 7 +1=15

Example 1: Finding the next terms in the sequence Given 𝑡 1 =5 𝑎𝑛𝑑 𝑡 𝑛 = 𝑡 𝑛−1 +3 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: #1-17odd