To find the Generating function for the given case

Question

$$a_{n} = \frac{4^{3n-5}}{3^{2n+4}}$$

I was just able to reach till $a_{n}$ = ($\frac{64}{9}$) $a_{n-1}$

Don't know how to proceed further

Answer 1

Using the formula for the sum of a geometric series: $$ \begin{align} \sum_{n=0}^\infty\frac{4^{3n-5}}{3^{2n+4}}x^n &=\sum_{n=0}^\infty\frac1{81\cdot1024}\left(\frac{64}9x\right)^n\\ &=\frac1{9\cdot9216}\frac1{1-\frac{64}9x}\\ &=\frac1{9216}\frac1{9-64x} \end{align} $$