SOLUTION: prove that. in a geometric series which has a sum to infinity. each term bears a constant ratio to the sum of all the following terms

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Question 982142: prove that. in a geometric series which has a sum to infinity. each term bears a constant ratio to the sum of all the following terms
Answer by Edwin McCravy(20054) About Me  (Show Source):
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prove that. in a geometric series which has a sum to infinity. each term bears a
constant ratio to the sum of all the following terms
We use S%5Binfinity%5D=a%2F%281-r%29 where "a" = the first term and r = the
common difference:

Suppose the series is 

The sum of all the terms following arn+1, using the formula is

ar%5E%28n-2%29%2F%281-r%29.

We want to show that the ratio of the nth term ar%5E%28n-1%29 to the sum of all
the following terms ar%5E%28n-2%29%2F%281-r%29 is a constant:

That ratio is found by dividing:

  ar%5E%28n-1%29%22%F7%22ar%5E%28n-2%29%2F%281-r%29 =

  ar%5E%28n-1%29%22%D7%22%281-r%29%2F%28ar%5E%28n-2%29%29 =

That simplifies to 

 r(1-r) which is a constant.

So we have proved the proposition.

Edwin