SOLUTION: The sum of the first two terms of an arithmetic progression is 23. The sum of the last two terms is -25. Find the sum of the progression if its first term is 13.

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Question 1092279: The sum of the first two terms of an arithmetic progression is 23. The sum of
the last two terms is -25. Find the sum of the progression if its first term is 13.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200) (Show Source):
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Given the first term is 13 and the sum of the first two terms is 23, we know the second term is 10. That means the common difference is -3.

Given that the sum of the last two terms is -25, and now knowing that the common difference is -3, we know the last two terms are -11 and -14.

The sum of all the terms is (number of terms) times (average of first and last terms). We know the first and last terms, so we can find their average. What we need to find is the number of terms.

The n-th term is the first term, plus the common difference (n-1) times. We can use this to determine the number of terms, n:

So there are 10 terms; and the average of the first and last terms is

And, finally, the sum of all the terms is

Answer by ikleyn(52792) (Show Source):
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.

The first term is 23 - 13 = 10. The common difference is -3. The equation for the last term, x, is x + (x+3) = -25, which implies 2x = -28, x = -14. So, the progression is 13, 10, 7, 4, 1, -2, -5, -8, -11, -14. 10 terms with the sum of -5.