SOLUTION: Consider the sequence where tn = 15 − 5n and the sum of the first m terms is −1250. Find the value of m.

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Question 1108766: Consider the sequence where tn = 15 − 5n and the sum of the first m terms is −1250.
Find the value of m.
Answer by greenestamps(13200) About Me  (Show Source):
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The sum of the first m terms is m, multiplied by the average of all the terms. In an arithmetic sequence, the average of all the terms is the average of the first and last terms. So
# terms = m
1st term: 15-5(1) = 10
m-th term: 15-5m


The sum of first m terms is -1250:
m%28%2810%2B%2815-5m%29%29%2F2%29+=+-1250
m%2825-5m%29+=+-2500
25m-5m%5E2+=+-2500
5m%5E2-25m-2500+=+0
m%5E2-5m-500+=+0
%28m-25%29%28m%2B20%29+=+0
m+=+25 or m+=+-20

Clearly only the positive solution to the equation makes sense in this problem.

The number of terms in the sequence is 25.