SOLUTION: The third term of a geometric sequence is 3 and the 6th term is 1/9? Find the first term.

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Question 1086032: The third term of a geometric sequence is 3 and the 6th term is 1/9? Find the first term.
Found 3 solutions by Fombitz, ikleyn, MathTherapy:
Answer by Fombitz(32388) About Me  (Show Source):
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a%5B3%5D=a%5B1%5D%2B%283-1%29d=a%5B1%5D%2B2d=3
a%5B6%5D=a%5B1%5D%2B%286-1%29d=a%5B1%5D%2B5d=1%2F9
Subtracting,
a%5B6%5D-a%5B3%5D=a%5B1%5D%2B5d-a%5B1%5D-2d=3d=1%2F9-3
3d=1%2F9-27%2F9
3d=-26%2F9
d=-26%2F27
So then,
a%5B1%5D%2B5%28-26%2F27%29=1%2F9
a%5B1%5D-130%2F27=3%2F27
a%5B1%5D=133%2F27

Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
.
a%5B3%5D = a%5B1%5D%2Ar%5E2 = 3,     (1)

a%5B6%5D = a%5B1%5D%2Ar%5E5 = 1%2F9.     (2)


Divide eq(2) by eq(1) (both sides). You will get 

a%5B6%5D%2Fa%5B3%5D = r%5E5%2Fr%5E2 = r%5E3 = %28%281%2F9%29%29%2F3 = 1%2F27,

which implies  

r = 1%2F3.


Thus we found the common ratio. It is r = 1%2F3.


Then from  (1)  a%5B1%5D = 3%2Fr%5E2 = 3%2F%28%281%2F9%29%29 = 27.


The progression is 27, 9, 3, 1, 1%2F3, 1%2F9.

Answer. a%5B1%5D = 27.

Solved.



Answer by MathTherapy(10551) About Me  (Show Source):
You can put this solution on YOUR website!

The third term of a geometric sequence is 3 and the 6th term is 1/9? Find the first term.
First term, or highlight_green%28matrix%281%2C3%2C+a%5B1%5D%2C+%22=%22%2C+27%29%29

IGNORE anyone who says otherwise!!