trigonometric series - Complex number trigonometry problem

Use $cos (n\theta)$ = $\frac{z^n +z^{-n}}{2}$ to express

$\cos \theta + \cos 3\theta + \cos5\theta + ... + \cos(2n-1)\theta$ as a geometric series in terms of z. Hence find this sum in terms of $\theta$.

I've tried everything in the world and still can't match that of the final answer. Could I pleas have a slight hint on the right path to follow.

Thanks