Solution:

$X\left(e^{j \omega}\right)=\sum_{n=0}^{L-1} A e^{-j \omega n}\\ $

$=A\left[\frac{1-e^{-j \omega L}}{1-e^{-j \omega}}\right]\\ $

$\sum_{n=0}^{N-1} x^n=\frac{1-x^N}{1-x}\\ $

$X\left(e^{j \omega}\right)=A\left[\frac{\left(e^{\frac{j \omega L}{2}}-e^{\frac{-j \omega L}{2}}\right) e^{\frac{-j \omega L}{2}}}{\left(e^{\frac{j \omega}{2}}-e^{-\frac{j \omega}{2}}\right) e^{\frac{-j \omega}{2}}}\right]\\ $

$=A\left[\frac{2 j \sin \frac{\omega L}{2}}{2 j \sin \frac{\omega}{2}}\right] e^{-\frac{j \omega(L-1)}{2}}\\ $

$ =A e^{-\frac{j \omega(L-1)}{2}}\left[\frac{\sin …