1. 12.5 Recursive Rules with Sequences & Functions
Alg II

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. Ex. 1: 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. Recursive Equations for…

6. So, a recursive rule would be a1=15, an=an-1+5
Ex. 2: Write the indicated rule for the arithmetic sequence with a1=15 and d=5.
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
Explicit rule
an=a1+(n-1)d
an=15+(n-1)5
an=15+5n-5
an=10+5n

7. So, a recursive rule for the sequence would be a1=4, an=0.2an-1
Ex. 3: Write the indicated rule for the geometric sequence with a1=4 and r=0.2.
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

8. 12-5B Recursive Rules with Sequences & functions

9. Iteration

11. Ex. 2 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

12. Ex. 3: 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)

13. Assignment