SOLUTION: what is the value of 9^(1/3) x 9^(1/9) x 9^(1/27) x ...? also is there a general formula in solving these types of question? thanks!

Algebra ->  Sequences-and-series -> SOLUTION: what is the value of 9^(1/3) x 9^(1/9) x 9^(1/27) x ...? also is there a general formula in solving these types of question? thanks!      Log On


   

Question 1107236: what is the value of 9^(1/3) x 9^(1/9) x 9^(1/27) x ...?
also is there a general formula in solving these types of question? thanks!
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
You know that %289%5Ex%29%289%5Ey%29%289%5Ez%29=9%5E%28%28x%2By%2Bz%29%29 .
So,
,
because
1%2F3%2B1%2F9%2B1%2F27%2B%22...%22=1%2F2

That sum is the sum of an infinite geometric sequence.
The "formula" for such a sum with a first term b and a common ratio r%3C1 is
sum=b%2F%281-r%29 .
In the case of the sum above, with b=1%2F3 and r=1%2F3 ,
sum=%221%2F3%22%2F%281-%221%2F3%22%29=%221%2F3%22%2F%222%2F3%22=1%2F2 .

There is no need to memorize formulas, if you can deduce them.
Consider the sum or n terms
S=b%2Bbr%2Bbr%5E2%2Bbr%5E3%2B%22...%22%2Bbr%5E%28n-1%29 .
rS=br%2Bbr%5E2%2Bbr%5E3%2B%22...%22%2Bbr%5E%28n-1%29%2Bbr%5En , and
rS-S=br%5En-b --> %28r-1%29S=b%28r%5En-1%29 --> S=b%28r%5En-1%29%2Fr-1 .
If r%3C1 , we might like to write it as
S=b%281-r%5En%29%2F%281-r%29 ,
and as n approaches infinity , r%5En approaches 0 ,
and S approaches b%2F%281-r%29 .