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Question 408537: The sum of the first two terms of a seven-term GP (geometric progression) is 20 and the sum of the last two terms is -5/8. Find the sum of all seven terms.
I know that a GP is one in which the ratio obtained by dividing any term by the preceding term is always the same. I also know the formula for finding the nth term in a geometric sequence. The first thing I tried was to let x = the first term. Then (20 - x) = the second term. Additionally, I let y = the next to the last term and (-5/8 - y) = the last term. Since the ratio by dividing any term by the preceding term is always the same, I have:
(20 - x) / x (ratio of 2nd term to 1st term)
(-5/8 - y) / y (ratio of 7th term to 6th term)
I set the equal to each other because the ratios are required to be the same in a GP:
(20 - x) / x = (-5/8 - y) / y
20y - xy = (-5/8)x - xy
20y = (-5/8)x
y = (-1/32)x
OR solving the same equation for x gives: x = -32y
From this point I thought about trying to just choose numbers either for x or y and then use the given common ratio to find the rest of the terms, but it didn't work out. Can you provide me with a step-by-step solution to this intriguing question? Thank you.
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! Maybe if you make use of the fact
that the 1st term is a, the 2nd term is ar
the 6th term is ar^5 and the 7th term
is ar^6, you can get a new relation of
x to y.
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Cheers,
Stan H.
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