I am having trouble simplifying the following expression:
$$ \frac{1}{7}\left(1+e^{-jk\frac{2\pi}{7}}+e^{-jk\frac{4\pi}{7}}+e^{-jk\frac{6\pi}{7}}+e^{-jk\frac{8\pi}{7}}\right) $$
I need to get $$ \frac{1}{7 \sin\left(\frac{\pi k}{7}\right)}\left(e^{-jk\frac{4\pi}{7}}\sin\left(\frac{5\pi k}{7}\right)\right) $$
Can someone guide me in the right direction? Thanks!
We sum the geometric progression of $e^{-i2\pi/7}$ to find that
$$\begin{align} \frac17\sum_{k=0}^4 \left(e^{-i2\pi/7}\right)^k&=\frac17\frac{1-e^{-i10\pi/7}}{1-e^{-i2\pi/7}}\\\\ &=\frac17\frac{e^{-i5\pi/7}(e^{i5\pi/7}-e^{-i5\pi/7})}{e^{-i\pi/7}(e^{i\pi/7}-e^{-i\pi/7})}\\\\ &=\frac17e^{-i4\pi/7}\frac{\sin(5\pi/7)}{\sin(\pi/7)} \end{align}$$