SOLUTION: An arithmetic series has a third term of 0. The sum of the first 15 terms is – 300. What is the first term?

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Question 692570: An arithmetic series has a third term of 0. The sum of the first 15 terms is – 300. What is the first term?
Found 2 solutions by mouk, MathTherapy:
Answer by mouk(232) About Me  (Show Source):
You can put this solution on YOUR website!
1st term = a, common difference = d
So terms are, a, a+d, a+2d, ...
3rd term = 0 means that +a%2B2d=0+ ............ Eq (1)
so +s%5Bn%5D=+%28n%2F2%29%282a+%2B+%28n-1%29d%29+
and +s%5B15%5D+=+%2815%2F2%29%282a%2B14d%29+=+-300+
+%2815%2F2%29%282a%2B14d%29+=+-300+
+2a%2B14d%29+=+-40+ ............ Eq (2)
Eq (2) - 7 Eq (1) gives:
+%282-7%29a+%2B%2814-14%29d=-40-0+
+-5a+=-40+
+a=8+, which is the first term

NB, If you need d then Eq (2) - 2 Eq (1) gives:
+%282-2%29a+%2B%2814-4%29d=-40-0+
+10d=-40+
+d=-4+

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

An arithmetic series has a third term of 0. The sum of the first 15 terms is – 300. What is the first term?

Formula for d, or the common difference in a series: d+=+%28a%5Bn%5D+-+a%5B1%5D%29%2F%28n+-+1%29, with a%5Bn%5D = a%5B3%5D = 0, and n = 3

Therefore, d+=+%28a%5Bn%5D+-+a%5B1%5D%29%2F%28n+-+1%29 becomes: d+=+%280+-+a%5B1%5D%29%2F%283+-+1%29 ------ d+=+%28-+a%5B1%5D%29%2F2 ------ eq (i)

Formula for sum of a series of numbers, or S%5Bn%5D: S%5Bn%5D+=+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29, with S%5Bn%5D = S%5B15%5D = - 300 ; and n = 15

Therefore, S%5Bn%5D+=+%28n%2F2%29%282a%5B1%5D+%2B+%28n+-+1%29d%29 becomes: S%5B15%5D+=+%2815%2F2%29%282a%5B1%5D+%2B+%2815+-+1%29d%29 --- -+300+=+7.5%282a%5B1%5D+%2B+14d%29 --- -+300+=+15a%5B1%5D+%2B+105d --- eq (ii)

-+300+=+15a%5B1%5D+%2B+105%28%28-+a%5B1%5D%29%2F2%29 ------ Substituting %28-+a%5B1%5D%29%2F2 for d in eq (ii)

-+300+=+15a%5B1%5D+%2B+52.5%28-+a%5B1%5D%29

-+300+=+15a%5B1%5D+-+52.5a%5B1%5D

-+37.5a%5B1%5D+=+-+300

a%5B1%5D, or 1st term = %28-+300%29%2F-+37.5, or highlight_green%288%29

You can do the check!!

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