So, I'm stuck with the algebra for the nth number on the Fibonacci sequence in here.
I managed to get to the part where $G(x) = \frac{x}{1-x-x^2}$ $=$ $\frac{x}{(1-\alpha x)(1-\beta x)}$, and I know that this can be written in the form of $\frac{A}{1-\alpha x} + \frac{B}{1-\beta x}$ which is going to get me to $\frac{\alpha^n - \beta^n}{\sqrt{5}}$, where $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta = \frac{1-\sqrt{5}}{2}$.
I just don't know how I can algebraically manipulate $\frac{x}{(1-\alpha x)(1-\beta x)}$ to $\frac{A}{1-\alpha x} + \frac{B}{1-\beta x}$.
Hopefully it's not something very obvious.
Sorry if bad english and if I assumed some wrong relation.
You actually do know how to manipulate $\frac{x}{(1-\varphi x)(1-\Phi x)}=\frac{A}{1-\varphi x}+\frac{B}{1-\Phi x}$, because all of the manipulations required are from elementary algebra: clear denominators and expand using the distributive property. Then you just have to equate coefficients (constant and linear terms).
This procedure generalizes in a big way; search for partial fraction decomposition.