Modulo arithmetic and sum of arbitrary powers of a primitive root of unity.

Question

Prove that if $w$ is a primitive nth root of unity, then

$1 + w^k + (w^k)^2 + (w^k)^3 + \cdots + (w^k)^{n-1} =0$ iff $k \neq 0$ mod $n$.

Sorry for the terrible formatting. Also, I don't know anything about groups and rings, so please keep the answer as elementary as possible! Thanks.

Note : $k$ is a non-zero integer.

Answer 1

Hint: $(1-x)(1+x+x^2+\ldots+x^{n-1}) = 1 - x^n$.