1. EXAMPLE 3 Write recursive rules for special sequences Write a recursive rule for the sequence. a. 1, 1, 2, 3, 5,...b. 1, 1, 2, 6, 24,... SOLUTION Beginning with the third term in the sequence, each term is the sum of the two previous terms. a. ANSWER So, a recursive rule is a 1 = 1, a 2 = 1, a n = a n – 2 + a n – 1.

2. EXAMPLE 3 Write recursive rules for special sequences So, a recursive rule is a 0 = 1, a n = n a n – 1. ANSWER b. Denote the first term by a 0 = 1. Then note that a 1 = 1 = 1 a 0, a 2 = 2 = 2 a 1, a 3 = 6 = 3 a 2, and so on.

3. EXAMPLE 4 Solve a multi-step problem Music Service An online music service initially has 50,000 annual members. Each year it loses 20% of its current members and adds 5000 new members. Write a recursive rule for the number a n of members at the start of the n th year..

4. EXAMPLE 4 Solve a multi-step problem Find the number of members at the start of the 5th year.. Describe what happens to the number of members over time.. SOLUTION Write a recursive rule. Because the number of members declines 20% each year, 80% of the members are retained from one year to the next. Also, 5000 new members are added each year. STEP 1

5. EXAMPLE 4 Solve a multi-step problem A recursive rule is a 1 = 50,000, a n = 0.8a n – 1 + 5000. ANSWER

6. EXAMPLE 4 Solve a multi-step problem STEP 2 Find the number of members at the start of the 5th year. Enter 50,000 (the value of a 1 ) into a graphing calculator. Then enter the rule 0.8 Ans + 5000 to find a 2. Press three more times to find a 5.

7. EXAMPLE 4 Solve a multi-step problem STEP 3 Describe what happens to the number of members over time. Continue pressing on the calculator. As shown at the right, after many years the number of members approaches 25,000.

8. GUIDED PRACTICE for Examples 3 and 4 9. Write a recursive rule for the sequence. a. 1, 2, 2, 4, 8, 32,.... ANSWER So, a recursive rule is a 1 = 1, a 2 = 2, a n = (a n – 2 ) (a n – 1 ).

9. GUIDED PRACTICE for Examples 3 and 4 10. WHAT IF? In Example 4, suppose 70% of the member are retained each year. What happen to the number of member over time? The number of members stabilizes at about 16,667 members. ANSWER