I really need some help in doing this:
By using the generating functions $F(z)$ and $L(z)$ for Fibonacci and Lucas numbers, show that: $$ F_n = \frac{L_{n-1}}{2}+\frac{L_{n-2}}{2^2}+\ldots+\frac{L_0}{2^n}.$$
I have found that $F(z)=\frac{z}{1-z-z^2}$ and $L(z) = \frac{2-z}{1-z-z^2}$ but I am stuck here.
Any assistance would be helpful.