EXAMPLE 3 Write recursive rules for special sequences Write a recursive rule for the sequence. a. 1,...
9
EXAMPLE 3 Write recursive rules for special sequences Write a recursive rule for the sequence. 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 a recursive rule is a 1 = 1, a 2 = 1, a n = a n – 2 + a n – 1 .
-
Upload
jack-griffin -
Category
Documents
-
view
216 -
download
0
Transcript of EXAMPLE 3 Write recursive rules for special sequences Write a recursive rule for the sequence. a. 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 a1 = 1, a2 = 1, an = an – 2 + an – 1.
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.
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.
.
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
EXAMPLE 4 Solve a multi-step problem
A recursive rule is a1 = 50,000, an = 0.8an – 1 + 5000.
ANSWER
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.
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.
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).
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
