sequences and series - How to find the sum of a geometric progression involving cos using complex numbers?

Use $2\cos{n\theta} = z^n + z^{-n}$ to express $\cos\theta + \cos3\theta + \cos5\theta + ... + \cos(2n-1)\theta$ as a geometric progression in terms of $z$. Hence find the sum of this progression in terms of $\theta$. Any tips/help would be appreciated. I have the common ratio as $z^2$ and the first term as $z^{1-2n},$ and I can put these into the original formula, but I can't seem to get the answer I'm looking for.

z= $\cos\theta + i\sin\theta$ where i is the imaginary unit