If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, an

Max@Math Revolution wrote:[GMAT math practice question]

If the sequence {An } satisfies An = An-1 - An-2, A1 = 0, and A2 = 1, where n is an integer greater than 2, then what is the sum of the first 100 terms of {An }?

A. 1
B. 2
C. 3
D. 4
E. 5

Let’s list the first few terms to discern a pattern.

A1 = 0
A2 = 1
A3 = 1 - 0 = 1
A4 = 1 - 1 = 0
A5 = 0 - 1 = -1
A6 = -1 - 0 = -1
A7 = -1 - (-1) = 0
A8 = 0 - (-1) = 1
A9 = 1 - 0 = 1

At this point, we can see that the terms repeat themselves in a cycle of 6 numbers: 0, 1, 1, 0, -1, -1 (notice that A7 = A1, A8 = A2, A9 = A3, etc.). Also notice that the sum of the 6 numbers in one cycle is 0. So the sum of all the terms up to and including the 96th term is 0 (notice 96 = 6 x 16). So we really just need to add A97, A98, A99 and A 100. Since A97 = A1 = 0, A98 = A2 = 1, A99 = 1 and A100 = 0, the sum of these 4 terms (and hence the sum of the first 100 terms) is 0 + 1 + 1 + 0 = 2.

Answer: B