Answer: (C) any odd number
Since $(-1)^8 = 1$, the equation becomes $(-1)^n + 1 = 0$, so $(-1)^n = -1$, which holds when $n$ is any odd number.
$(-1)^{\text{even}} = +1$ and $(-1)^{\text{odd}} = -1$. Since $(-1)^8 = 1$, we need $(-1)^n = -1$, which requires $n$ to be odd. Note: $n$ need not be positive or negative specifically — any odd integer works, making (C) the correct and most precise option.