Question 1130438: the sum of the first 100 terms of the sequence 1,-2,3,4,-5,6,7,-8,9,10... is 1750. what is the sum of the first 100 terms in the sequence 1,2,-3,4,5,-6,7,8,-9,1o?
Answer by greenestamps(13206)   (Show Source):
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It is relatively easy to find the sum of the first 100 terms of the second sequence without knowing anything about the first sequence. However, the way the problem is written, we are apparently supposed to find the sum of the first 100 terms of the second sequence by comparing the two sequences.
We can do this, knowing that the sum of the first 100 terms of the first sequence is 1750.
Grouping the first 99 terms the first sequence in groups of 3, we get the following sequence of 33 terms:
2, 5, 8, ..., 95, 98
Grouping the first 99 terms the second sequence in groups of 3, we get the following sequence of 33 terms:
0, 3, 6, ..., 93, 96
Each term of the first of these new sequences is 2 more than the corresponding term of the new second sequence; since there are 33 terms in each new sequence, the difference between the sums of the two new sequences is 33*2 = 66.
The 100th term of both original sequences is 100, so the difference between the sums of the two original sequences is 66.
So the sum of the second sequence is 1750-66 = 1684.
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