How would you show that $\displaystyle \sum_{k=-n}^n e^{ikx} =\frac{\sin\left(\left(n+\frac{1}{2}\right)x\right )}{\sin(x/2)}$

WITHOUT using the method shown here...

Dirichlet kernel - Wikipedia, the free encyclopedia

Alternatively how would you prove that

$\displaystyle \sum_{k=-n}^n r^k=r^{-n}\cdot\frac{1-r^{2n+1}}{1-r}$?