How to find inverse z transform Suppose
$$Y(z) = \frac{\frac 12 z + 1}{z+\frac 12} \cdot \frac{z}{z-\frac12}\text.$$
According to Wolfram Alpha the inverse transform is, $2^{-n - 2} \cdot(5 - 3 \cdot (-1)^n)$. However, I cannot show why this is true. Could someone please guide me?
Using binomial formulas one easily gets
$$
Y(z)=\frac{\frac12+z^{-1}}{1-\frac14z^{-2}}
=(\tfrac12+z^{-1})\sum_{k=0}^\infty 2^{-2k}z^{-2k}.
$$
Now compare to the coefficients in $Y(z)=\sum_{n=0}^\infty y_nz^{-n}$ for $n=2k$ and $n=2k+1$, even and odd indices.