In this lesson you will learn another way to define a sequence — by a recursive rule .

1. Using Recursive Rules for Sequences So far you have worked with explicit rules for the nth term of a sequence, such as an = 3n – 2 and an = 3(2)n. An explicit rule gives an as a function of the term’s position number n in the sequence. In this lesson you will learn another way to define a sequence — by a recursive rule. A recursive rule gives the beginning term or terms of a sequence and then a recursive equation that tells how an is related to one or more preceding terms.

2. Evaluating Recursive Rules Write the first five terms of the sequence. Factorial numbers: a0 = 1, an = n•an– 1 SOLUTION a0 = 1 a1 = 1 •a0 = 1 • 1 = 1 a2 = 2 •a1 = 2 • 1 = 2 a3 = 3 •a2 = 3 • 2 = 6 a4 = 4 •a3 = 4 • 6 = 24

3. Evaluating Recursive Rules Write the first five terms of the sequence. Factorial numbers: a0 = 1, an = n•an– 1 Fibonacci sequence: a1 = 1, a2= 1, an = an– 2 + an– 1 SOLUTION a0 = 1 a1 = 1 a1 = 1 •a0 = 1 • 1 = 1 a2 = 1 = a1+ a2 = 1 + 1 a2 = 2 •a1 = 2 • 1 = 2 a3 = 2 = a2+ a3 = 1 + 2 a3 = 3 •a2 = 3 • 2 = 6 a4 = 3 = a3+ a4 = 2 + 3 a4 = 4 •a3 = 4 • 6 = 24 a5 = 5

4. Evaluating Recursive Rules Factorial numbers are denoted by a special symbol, !, called a factorial symbol. The expression n! is read “n factorial” and represents the product of all integers from 1 to n. Here are several factorial values. 0! = 1 (by definition) 1! = 1 2! = 2 • 1 = 2 3! = 3 • 2 • 1 = 6 4! = 4 • 3 • 2 • 1 = 24 5! = 5 • 4 • 3 • 2 • 1 = 120

5. Evaluating Recursive Rules ACTIVITY Developing Concepts 1 2 INVESTIGATING RECURSIVE RULES Find the first five terms of each sequence. a1 = 3 a1= 3 an = an– 1 + 5 an= 2an– 1 Based on the lists of terms you found in Step 1, what type of sequence is the first recursive rule? the second recursive rule?

6. Writing a Recursive Rule for an Arithmetic Sequence 4 3 Write the indicated rule for the arithmetic sequence with a1= 4 and d = 3. an explicit rule SOLUTION From a previous lesson you know that an explicit rule for the nth term of the arithmetic sequence is: an = a1 + (n – 1)d General explicit rule for an = a1 + (n – 1)d Substitute fora1 and d. = 1 + 3n Simplify.

7. Writing a Recursive Rule for an Arithmetic Sequence Write the indicated rule for the arithmetic sequence with a1 = 4 and d = 3. a recursive rule SOLUTION To find the recursive equation, use the fact that you can obtain an by adding the common difference d to the previous term. an = an– 1 + d General recursive rule for an = an – 1 + d 3 Substitute ford. A recursive rule for the sequence is a1 = 4, an = an– 1 + 3.

8. Writing a Recursive Rule for a Geometric Sequence Write the indicated rule for the geometric sequence with a1 = 3 and r = 0.1. an explicit rule SOLUTION From previous lesson you know that an explicit rule for the nth term of the geometric sequence is: an = a1 rn – 1 General explicit rule for an = a1rn – 1 3(0.1)n – 1 Substitute fora1 and r.

9. Writing a Recursive Rule for a Geometric Sequence Write the indicated rule for the geometric sequence with a1 = 3 and r = 0.1. a recursive rule SOLUTION To write a recursive rule, use the fact that you can obtain an by multiplying the previous term by r. an = r•an– 1 General recursive rule for an = r•an– 1 Substitute forr. (0.1) A recursive rule for the sequence is a1 = 3, an = (0.1)an– 1.

10. Writing a Recursive Rule Write a recursive rule for the sequence 1, 2, 2, 4, 8, 32, … SOLUTION Beginning with the third term in the sequence, each term is the product of the two previous terms. Therefore, a recursive rule is given by: a1 = 1, a2 = 2, an = an– 2•an – 1

11. Using Recursive Rules in Real Life Fish at start of nth year Fish at start of (n – 1)st year New fish added = 0.7 + FISH A lake initially contains 5200 fish. Each year the population declines 30% due to fishing and other causes, and the lake is restocked with 400 fish. Write a recursive rule for the number an of fish at the beginning of the nth year. How many fish are in the lake at the beginning of the fifth year? SOLUTION Because the population declines 30% each year, 70% of the fish remain in the lake from one year to the next, and new fish are added. Verbal Model Fish at start of nth year = an Labels Fish at start of (n – 1)st year = an – 1 New fish added = 400 an = (0.7)an – 1 + 400 Algebraic Model

12. Using Recursive Rules in Real Life a4 a3 There are about 2262 fish in the lake at the beginning of the fifth year. a2 a1 FISH A lake initially contains 5200 fish. Each year the population declines 30% due to fishing and other causes, and the lake is restocked with 400 fish. Write a recursive rule for the number an of fish at the beginning of the nth year. How many fish are in the lake at the beginning of the fifth year? SOLUTION a1 = 5200, an = (0.7)an – 1 + 400 A recursive rule is: 2659.6  2262 Find a5: = (0.7) an – 1 + 400 = 2261.72 a5 = (0.7)an – 1 + 400 a4 3228 = 2659.6 = (0.7)an – 1 + 400 a3 = 3228 4040 = (0.7)an – 1 + 400 a2 5200 = 4040

13. Using Recursive Rules in Real Life Over time, the population of fish in the lake stabilizes at about 1333 fish. FISH A lake initially contains 5200 fish. Each year the population declines 30% due to fishing and other causes, and the lake is restocked with 400 fish. What happens to the population of fish in the lake over time? SOLUTION You can use a graphing calculator to determine what happens to the lake’s fish population over time. Observe that the numbers approach about 1333 as n gets larger.