Multiplication proof by induction

Question

Prove by induction that $$\sum_{r=0}^{n-1} r = \frac{1}{2}n(n-1)$$

For all $n\in\mathbb{N}$.

This is straight forward. But how can it be used to show the following:

$$\Pi_{r=0}^{n-1} e^\frac{2ri\pi}{n}=(-1)^{n-1}$$

For all $n\in\mathbb{N}$.

Answer 1

$\prod{e^\frac{2i\pi r}{n}}= \exp( \frac{2i \pi \sum{r}}{n}) = \exp ( \frac{ 2i \pi n (n -1)}{2n}) = (e^{i \pi})^{n-1}=(-1)^{n-1}$