SOLUTION: An arithmetic series has the following properties (i) the sum of the fourth and ninth terms is 58. (ii) the sum of the first 26 terms is 390. (a) Find the first term and the com

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Question 1136301: An arithmetic series has the following properties
(i) the sum of the fourth and ninth terms is 58.
(ii) the sum of the first 26 terms is 390.
(a) Find the first term and the common difference.
(b) Find the smallest integer value of n for which the sum to n terms of the series is negative.
Answer by greenestamps(13203) About Me  (Show Source):
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Let a = first term
let d = common difference

Then 4th term = a+3d
and 9th term = a+8d

Sum of 4th and 9th terms = 58 =(a+3d)+(a+8d)

2a%2B11d+=+58 (1)

Sum of first 26 terms = 390 = 26*average of 1st and 26th terms

26%28%28a%2Ba%2B25d%29%2F2%29=390
2a%2B25d+=+30 (2)

From (1) and (2)...

2a%2B25d+=+30
2a%2B11d+=+58
14d+=+-28
d+=+-2

Then

2a%2B11%28-2%29+=+58
2a-22+=+58
2a+=+80
a+=+40

The first term is 40; the common difference is -2.

With first term 40 and common difference -2, the 21st term will be 40-40 = 0 and the 41st term will be 40-80 = -40. That means the sum of the first 81 terms will be 0, because all the positive terms at the beginning of the sequence get cancelled out by later negative terms.

So the smallest n for which the sum of n terms of the sequence is negative is 82.