EXAMPLE 3 Write recursive rules for special sequences Write a recursive rule for the sequence. a. 1,...
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Transcript of EXAMPLE 3 Write recursive rules for special sequences Write a recursive rule for the sequence. a. 1,...
![Page 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/1.jpg)
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 a1 = 1, a2 = 1, an = an – 2 + an – 1.
![Page 2: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/2.jpg)
EXAMPLE 3 Write recursive rules for special sequences
So, a recursive rule is a0 = 1, an = n an – 1.
ANSWER
b. Denote the first term by a0 = 1. Then note that a1 = 1 = 1 a0, a2 = 2 = 2 a1, a3 = 6 = 3 a2, and so on.
![Page 3: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/3.jpg)
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 an of members at the start of the nth year.
.
![Page 4: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/4.jpg)
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
![Page 5: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/5.jpg)
EXAMPLE 4 Solve a multi-step problem
A recursive rule is a1 = 50,000, an = 0.8an – 1 + 5000.
ANSWER
![Page 6: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/6.jpg)
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 a1) into a graphing calculator. Then enter the rule 0.8 Ans + 5000 to find a2. Press three more times to find a5.
![Page 7: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/7.jpg)
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.
![Page 8: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/8.jpg)
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 a1 = 1, a2 = 2, an = (an – 2) (an – 1).
![Page 9: 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.](https://reader036.fdocuments.us/reader036/viewer/2022082518/5697bf9e1a28abf838c9427c/html5/thumbnails/9.jpg)
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