Question 1108766
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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
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The sum of first m terms is -1250:
{{{m((10+(15-5m))/2) = -1250}}}
{{{m(25-5m) = -2500}}}
{{{25m-5m^2 = -2500}}}
{{{5m^2-25m-2500 = 0}}}
{{{m^2-5m-500 = 0}}}
{{{(m-25)(m+20) = 0}}}
{{{m = 25}}} or {{{m = -20}}}<br>
Clearly only the positive solution to the equation makes sense in this problem.<br>
The number of terms in the sequence is 25.