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

9. 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

10. 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

11. 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?

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

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

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

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

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