11.5 Recursive Rules for Sequences

1. 11.5 Recursive Rules for Sequences p. 681

2. Explicit Rule • A function based on a term’s position, n, in a sequence. • All the rules for the nth term that we’ve been working with are explicit rules; such as an=a1rn-1.

3. Recursive Rule • Gives the first term(s) of a sequence and an equation that relates the given term(s) to the next terms in the sequence. • For example: Given a0=1 and an=an-1-2 • The 1st five terms of this sequence would be: a0, a1, a2, a3, a4 OR • 1, -1, -3, -5, -7

4. Example: Write the 1st 5 terms of the sequence. • a1=2, a2=2, an=an-2-an-1 a3=a3-2-a3-1=a1-a2=2-2=0 a4=a4-2-a4-1=a2-a3=2-0=2 a5=a5-2-a5-1=a3-a4=0-2=-2 2, 2, 0, 2, -2 2nd term 1st term

5. Explicit rule an=a1+(n-1)d an=15+(n-1)5 an=15+5n-5 an=10+5n Recursive rule (*Use the idea that you get the next term by adding 5 to the previous term.) Or an=an-1+5 So, a recursive rule would be a1=15, an=an-1+5 Example: Write the indicated rule for the arithmetic sequence with a1=15 and d=5.

6. Explicit rule an=a1rn-1 an=4(0.2)n-1 Recursive rule (*Use the idea that you get the next term by multiplying the previous term by 0.2) Or an=r*an-1=0.2an-1 So, a recursive rule for the sequence would be a1=4, an=0.2an-1 Example: Write the indicated rule for the geometric sequence with a1=4 and r=0.2.

7. Write a recursive rule for the sequence 1,2,2,4,8,32,… . • First, notice the sequence is neither arithmetic nor geometric. • So, try to find the pattern. • Notice each term is the product of the previous 2 terms. • Or, an-1*an-2 • So, a recursive rule would be: a1=1, a2=2, an= an-1*an-2

8. Example: Write a recursive rule for the sequence 1,1,4,10,28,76. • Is the sequence arithmetic, geometric, or neither? • Find the pattern. • 2 times the sum of the previous 2 terms • Or 2(an-1+an-2) • So the recursive rule would be: a1=1, a2=1, an= 2(an-1+an-2)

9. Assignment