Question 1161936: Find a general term in simplest form for the sequence:2,1,-4,7,-10,13,-16,...
Answer by greenestamps(13200)   (Show Source):
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2, 1, -4, 7, -10, 13, -16
The signs are alternating (except at the beginning). To analyze the sequence, change the sign of every other term to make most of the terms positive:
-2, 1, 4, 7, 10, 13, 16
We see this is an arithmetic sequence with common difference 3 and first term -2. That makes the formula for the n-th term
t(n) = 3n-5
To get the alternating signs, we can add a factor of (-1)^n or (-1)^(n+1). If we add a factor of (-1)^n, then the signs of the odd-numbered terms change, because -1 to any odd power is -1.
That gives us the desired sequence:
2, 1, -4, 7, -10, 13, -16
So the formula for the general term is
t(n) = ((-1)^n)(3n-5)
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