SOLUTION: The sum of the terms of an infinite geometric sequence is 18 and the sum of the first two terms of the same sequence is 10. If some of the terms of the sequence are negative, what

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Question 773298: The sum of the terms of an infinite geometric sequence is 18 and the sum of the first two terms of the same sequence is 10. If some of the terms of the sequence are negative, what is the seventh term of the sequence?

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Answer by reviewermath(1029) About Me  (Show Source):
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Q:
The sum of the terms of an infinite geometric sequence is 18 and the sum of the first two terms of the same sequence is 10. If some of the terms of the sequence are negative, what is the seventh term of the sequence?
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A:
S = a%5B1%5D%2F%281+-+r%29 = 18
a%5B1%5D = 18(1 - r)
a%5B1%5D + a%5B1%5Dr = 10
18(1 - r) + 18(1 - r)r = 10
9r%5E2+-+4+=+0
r = -sqrt%284%2F9%29 = -2%2F3
We choose the negative root because some terms are negative.
a%5B1%5D = 18%281+-+%28-2%2F3%29%29 = 30
a%5B7%5D = a%5B1%5Dr%5E6 = %2830%29%28-2%2F3%29%5E6 = highlight%28640%2F243%29