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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

If a sequence is defined by:

what is the sum of 2010 consecutive terms?

Notice that the exponent is always a multiple of 3.

Consecutive multiples of 3 are odd, even, odd, even, etc.

Since 2010 is even, no matter at what rank we start the sequence, half the terms are equal to -1 and the other half are equal to 1. The sum is 0.