Im confused i tried the power thing but how do you compute at the end

Because $(i, i^2, i^3, i^4) = (i, -1, -i, 1)$, we have  $i + i^2 + i^3 \cdots + i^{258} + i^{259} = 64(i - 1 - i + 1) + i^{257} + i^{258} + i^{259}$. So, the desired sum is $i^{257} + i^{258} + i^{259} = i - 1 - i = \boxed{-1}$.