If $S_n=81+54+36+24+\dots$ upto n terms, then value of 

$x=\frac{S_n-4S_{n-1}+6S_{n-2}-4S_{n-3}+S_{n-4}}{S_{n-1}-4S_{n-2}+6S_{n-3}-4S_{n-4}+S_{n-5}}$ is equal to 

Here, $a=81,r=\frac{2}{3}$

Also , $S_n-S_{n-1}=a{r}^{n-1}$

Let$y =$ numerator of $x$, then

$y=\left(S_n-S_{n-1}\right)-3\left(S_{n-1}-S_{n-2}\right)$

$+3\left(S_{n-2}-S_{n-3}\right)-\left(S_{n-3}-S_{n-4}\right)$

$=a{r}^{n-1}-3a{r}^{n-2}+3a{r}^{n-3}-a{r}^{n-4}$

$=a{r}^{n-4}\left(r^3-3r^2+3r-1\right)$

Thus , $x=r=\frac{2}{3}$