Edit: I actually found was I was missing. I forgot to use the right shift rule after deriving from $\frac{1}{1-x}$, as such my $\frac{\alpha}{(1-x)^2}$ should've been a $\frac{\alpha x}{(1-x)^2}$ for $a_n = n$
How to find the generating function for the sequence $a_n = \alpha n + \beta$ ?
I started by isolating $a_n = \alpha n$ and $b_n = \beta$, for which I found respectively $ \frac{\alpha}{(1-x)^2}$ and $\frac{\beta}{1-x}$
My idea was if I added them together, I should have the appropriate generating function, but apparently I'm missing something?
Thank you in advance!