Answer

If we ignore the signs we see the sequence

1, 2, 4, 8, 16,...

These are the powers of 2 starting at n = 0. To account for the signs we need to multiply by (-1)^{n + 1}. Starting at n = 0, the general term is

a_n = (-1)^{n + 1}2ⁿ.

If we start instead at n = 1 the general term is

a_n = (-1)ⁿ2^{n – 1}.