Inverse Z transform of $\dfrac{2(z^2-5z+6.5)}{(z-2)(z-3)}$, for $2<\left|z\right|<3$

Question

I want to find inverse $\mathcal{Z}$ transform of

$\dfrac{2(z^2-5z+6.5)}{(z-2)(z-3)}$ valid on an annulus region for example for $2<\left|z\right|<3$

Answer 1

$$\dfrac{2(z^2-5z+6.5)}{(z-2)(z-3)}=2-\frac{1}{z-2}+\frac{1}{z-3}$$ we can make it more clear as $$2-\frac{z^{-1}}{1-2z^{-1}}+\frac{z^{-1}}{1-3z^{-1}}$$

Now consider the given ROC, and the table of inverse $z$-transforms, and notice that $\mathcal{Z}^{-1}\{z^{-k}X(z)\}=x[n-k]$ to find the inverse $z$-transform.