I have done the first part of this question and proved the statement is true by induction. But I am not sure about the second part.
Use induction on $n$ to erify that $1 + z + \cdots + z^n = \frac{1-z^{n+1}}{1-z}\quad\text{(for }z\neq 1\text{)}$ Use this to show that if $c$ is an $n$th root of $1$ and $c\neq 1$, then $1+c+\cdots + c^n = 0$.