SOLUTION: The second term and the fifth term of a geometric series are 3 and 81 respectively. Find the sum from the fifth term to the tenth term of this series.

Question 1070873: The second term and the fifth term of a geometric series are 3 and 81 respectively. Find the sum from the fifth term to the tenth term of this series.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!

Mental math says that the common ratio is .
With formulas, id is the first term,
and is the common ratio,
, so
is , the fifth term,
is , the second term,
,
, and substituting
,
,
, and .
The sum of terms starting from a term
in a geometric series with common ratio , is
.
We are skipping the first terms of a geometric series with ,
and adding up the next terms,
starting from fifth term and adding up to the tenth term,
so the sum is

You may think that I need to find the first term of that series,
then calculate the sum of the first 10 terms,
next calculate the sum of the first 4 terms,
and finally subtract one sum from the other.
That is inefficient.
Who can say that I am not allowed to think of my own series,
which happens to have the same as the series in the problem,
and the same terms, except for missing the first 4?
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