I am doing a review of stuff from earlier in the semester and I can't prove this by induction:
Use induction on $n$ to verify that $1+x+\cdots+z^n= \frac{1-z^{n+1}}{1-z}$ (for $z\not=1)$. Use this to show that if $c$ is an $n$-th root of $1$ and $c\not=1$, then $1+c+\cdots+c^n=0$.
There is also a follow up question based on that one:
Show that if $c$ is any $n$-th root of $1$ and $c\not=1$ then $1+c+c^2+\cdots+c^{n-1}=0$
Note: If memory serves me correctly there is a misprint in one of these questions. I can't remember which one.