Question 1092277: if S10 = 50, S50 = 0 then what is a10, S100 (Geometric Sequence)
Answer by greenestamps(13200)   (Show Source):
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Using a for the first term and d for the common difference...
S(10) = a + (a+d) + (a+2d) + ... + (a+9d) = 10a+45d
S(50) = a + (a+d) + (a+2d) + ... + (a+49d) = 50a+1225d
So we know
Multiply the first equation by 5 and subtract from the second equation to eliminate variable a, and solve for d.
The common difference, d, is -1/4. Plug this value into an earlier equation to solve for a.
The first term, a, is 49/8. The common difference, d, is -1/4.
We are asked to find the 10th term, and the sum of the first 100 terms.
The 10th term is the first term, plus the common difference 9 times:
The 10th term is 31/8.
The sum of the first 100 terms is 100 times the average of the first and last (100th) terms. The 100th term is the first term, plus the common difference 99 times:
The average of the first and 100th terms is
And finally the sum of the first 100 terms is
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