8.5 Using Recursive Rules with Sequences

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

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

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

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

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…

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

Example 4 Recursive equation for arithmetic sequence Substitute 8 for d

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

Example 6

Example 7