SOLUTION: The sum of an arithmetic series is 1356 and the first term is -1. If the common difference is 5, how many terms are in the sequence?

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Question 336261: The sum of an arithmetic series is 1356 and the first term is -1. If the common difference is 5, how many terms are in the sequence?

Answer by jrfrunner(365) About Me  (Show Source):
You can put this solution on YOUR website!
arithmetic series: a%5B1%5D+%2B+%28a%5B1%5D%2Bdiff%29%2B%28a%5B1%5D%2B2%2Adiff%29 +...+ %28a%5B1%5D%2B%28n-1%29%2Adiff%29=sum%28%28a%5B1%5D%2B%28i-1%29%2Adiff%29%29%29 summed from i=1 to n
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Given a%5B1%5D=-1, diff=5 and sum=1356
sum%28%28a%5B1%5D%2B%28i-1%29%2Adiff%29%29=sum%28%28-1%2B%28i-1%29%2A5%29%29=1356
sum%28%285i-6%29%29=1356
5%2Asum%28i%29-n%2A6=1356
5%2An%2A%28n%2B1%29%2F2-6%2An=1356
5%2An%2A%28n-1%29-12%2An=2712 (multiply both sides by 2)
5n%5E2%2B5%2An-12%2An=2712
5%2An%5E2-7%2An-2712=0
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using quadratic solution
with a=5, b=-7, c=-2712
n=%287-Sqrt%287%5E2-4%2A%285%29%2A%28-2712%29%29%29%2F%282%2A5%29=%287-233%29%2F10=-22.6
or
n=%287%2BSqrt%287%5E2-4%2A%285%29%2A%28-2712%29%29%29%2F%282%2A5%29=%287%2B233%29%2F10=24
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since we need the solution to be positive, the first solution is not needed
and n=24