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Recursive Rules for Special Sequences

Learn how to write recursive rules for special sequences using examples. Also, solve a multi-step problem and find the number of members over time.

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Recursive Rules for Special Sequences

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  1. a. Beginning with the third term in the sequence, each term is the sum of the two previous terms. ANSWER So, a recursive rule is a1 = 1, a2 = 1, an= an – 2 + an–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

  2. b.Denote the first term by a0 = 1. Then note that a1= 1= 1a0, a2= 2 = 2a1, a3= 6 = 3a2, and so on. ANSWER So, a recursive rule isa0 = 1, an= n an – 1. EXAMPLE 3 Write recursive rules for special sequences

  3. . Write a recursive rule for the number anof members at the start of the nth year. 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.

  4. . Find the number of members at the start of the 5th year. . Describe what happens to the number of members over time. EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 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.

  5. ANSWER A recursive rule isa1 = 50,000, an= 0.8an– 1 + 5000. EXAMPLE 4 Solve a multi-step problem

  6. Find the number of members at the start of the 5th year. Enter 50,000 (the value of a1) into a graphing calculator. Then enter the rule 0.8 Ans+ 5000to finda2. Press three more times to finda5. EXAMPLE 4 Solve a multi-step problem STEP 2

  7. 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. EXAMPLE 4 Solve a multi-step problem STEP 3 Describe

  8. ANSWER So, a recursive rule is a1 = 1, a2 = 2, an = (an – 2) (an – 1). for Examples 3 and 4 GUIDED PRACTICE 9. Write a recursive rule for the sequence. a. 1, 2, 2, 4, 8, 32, . . . .

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

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