I know that stack exchange mathematics is not a site for asking question answer but for the sake of explaining the question I need to upload this question
Is there any way that we can find the solution of this question by putting values in the answer and checking them? Or in other words can we find the solution from the options given rather than solving the generating function?
Try $z=\frac{1}{9}$. Then
$\sum_{r=0}^{\infty}a_rz^r=\sum_{r=0}^{\infty}3^{2+r}3^{-2r}=\sum_{r=0}^{\infty}3^{2-r}=9\sum_{r=0}^{\infty}3^{-r}=\frac{27}{2}$
and I think only one of the answer choices matches that value.
(Although writing down an expression for $\sum_{r=0}^{\infty}3^{r+2}z^r=9\sum_{r=0}^{\infty}(3z)^r$ is just as easy)