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Question 477818: The sum of the first two terms of a geometric sequence is 90. The sum of the sixth and seventh terms is -10/27. Find the sum of the first seven terms of the sequence.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! The sum of the first two terms of a geometric sequence is 90. The sum of the sixth and seventh terms is -10/27. Find the sum of the first seven terms of the sequence.
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I could not get the problem to work if the sum of the 6th and 7th terms is -10/27. If I assume the sum is actually 10/27, then the problem can be solved.
For a geometric sequence, the nth term can be written as:
a_n = ar^(n-1) where a = the 1st term, r = the common ratio
Then the sum of the 1st two terms can be written
90 = a + ar = a(1+r) [1]
The sum of the 6th and 7th terms can be written
10/27 = ar^5 + ar^6 = ar^5(1+r) [2]
From [1], 1+r = 90/a
Substitute this value into [2]:
10/27 = ar^5(90/a) = 90r^5
Solve for r:
r^5 = 10/(27*90)
r = (10(27*90))^(1/5)
This gives r = 1/3
Now we can solve for a:
90 = a(1+1/3) = a(4/3) -> a = 67.5
The sum of the first n terms of a geometric series is:
S_n = a(1-r^n)/(1-r)
So the sum of the first 7 terms is
So S_7 = 67.5(1-(1/3)^7)/(1-1/3) = 101.204
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