1. 8.5 Using Recursive Rules with Sequences

2. Explicit Rule of Sequence
The explicit rule of a sequence gives an as a function of the term’s position number n in the sequence.
For example

3. Recursive Rule of Sequence
The recursive rule of a sequence is another way to define a sequence. It gives the beginning term(s) of a sequence and a recursive equation that tells how an is related to one or more preceding terms.
For example
1st term
Given the recursive rule:
Recursive Equation
Let’s use the recursive rule to find the first 6 terms of the sequence
1st term
4th term
2nd term
5th term
3rd term
6th term

4. Example 1 Write the first six terms of the sequence.
1st term
Recursive Equation
1st term
2nd term
3rd term
4th term
5th term
6th term

5. Example 2 Write a recursive rule for 16, 40, 100, 250, 625… 1 2 3 4 5
Recursive equation for geometric sequence
Substitute for r

6. Example 3 Write a recursive rule for 1, 1, 2, 6, 24…
The terms have neither a common difference nor a common ratio.
First term:
Note that
and so on…

7. Recursive vs. Explicit Recursive Definition Explicit Definition
Compare Recursive vs. Explicit:
Recursive Definition
Explicit Definition

8. Example 4 Recursive equation for arithmetic sequence
Substitute 8 for d

9. Example 5 Explicit rule for geometric sequence
Substitute 10 for a1 and 2 for r

10. Example 6

11. Example 7