SOLUTION: The 3rd term of a geometric sequence is 36, and the 6th term is 9/2. What is the recursive formula for the sequence?

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Question 1202469: The 3rd term of a geometric sequence is 36, and the 6th term is 9/2. What is the recursive formula for the sequence?
Found 3 solutions by mananth, ikleyn, greenestamps:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
The 3rd term of a geometric sequence is 36, and the 6th term is 9/2. What is the recursive formula for the sequence

tn = ar^(n-1)
t3= a*r^(3-1)
t3= ar^2=36
t6 = a*r(6-1) = ar^5= 9/2
t3/t6 = ar^2/ar^5 = 36/(9/2)
1/r^3= 36/9 *2
1/r^3= 8
r^3= 1/8
r=1/2
ar^2=36
plug r
a*(1/2)^2 =36
a/4 =36
a= 144
tn = 144*(1/2)^n-1



Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.

D E F I N I T I O N


    +--------------------------------------------------------------+
    |  A recursive formula is a formula that defines               |
    |  any term of a sequence in terms of its preceding term(s).   |
    +---------------------------------------------------------------+



In this problem, the recursive formula is 

    a%5Bn%2B1%5D = %281%2F2%29%2Aa%5Bn%5D,  a%5B1%5D = 144,  n = 1, 2, 3, . . .

Solved.


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To answer correctly, the reader must understand the difference
between conceptions of explicit formula and recursive formula.




Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The 3rd term, 36, is the first term, multiplied by the common ratio 2 times: ar%5E2=36.

The 6th term, 9/2, is the first term, multiplied by the common ratio 5 times: ar%5E5=9%2F2.

Divide the formulas for the 6th and 3rd terms to calculate the common ratio:

r%5E3=%289%2F2%29%2F36=1%2F8
r=1%2F2

A RECURSIVE formula tells how to get each term from the preceding term; for a geometric sequence the rule is "multiply by the common ratio". Since in this problem the common ratio is 1/2, the recursive formula is

a%28n%29=%281%2F2%29%2Aa%28n-1%29

To complete the definition of the recursive formula for the sequence, we need to specify the first term. Since the 3rd term is 36 and the common ratio is 1/2,

36=a%28%281%2F2%29%5E2%29
36=a%2F4
a=144

ANSWER: a(1)=144; for n>1, a(n)=(1/2)*a(n-1)