SOLUTION: In a geometric series with first term,
a
and common ratio,
r
(where
r
is real and
r 1
), the sum of
the first 7 terms is 4 times the sum of the following 7 terms. Find
Algebra ->
Sequences-and-series
-> SOLUTION: In a geometric series with first term,
a
and common ratio,
r
(where
r
is real and
r 1
), the sum of
the first 7 terms is 4 times the sum of the following 7 terms. Find
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Question 1184769: In a geometric series with first term,
a
and common ratio,
r
(where
r
is real and
r 1
), the sum of
the first 7 terms is 4 times the sum of the following 7 terms. Find the ratio of the sum of the first 21
terms to the sum of the first 14 terms. Answer by greenestamps(13200) (Show Source):
The sum of the first 7 terms is 4 times the sum of the following 7 terms.
In a geometric series, that means each term among the first 7 terms is 4 times a corresponding term among the following 7 terms.
Then, since the common ratio remains the same, the sum of the second set of 7 terms (terms 8 to 14) is 4 times the sum of the third set of 7 terms (terms 15 to 21).
So we can...
let x be the sum of terms 15-21
then 4x is the sum of terms 8-14
and 4(4x)= 16x is the sum of terms 1-7
So the sum of the first 21 terms is 21x and the sum of the first 14 terms is 20x. The ratio of the sum of the first 21 terms to the sum of the first 14 terms is then 21x/20x = 21/20.
ANSWER: 21/20, or 21:20
Note the given information tells us that the common ratio in the sequence is
Fortunately, we don't need to deal with that number in this problem, because the problem involves groups of 7 consecutive terms of the sequence.
