In a geometric progression $\left\{a_n\right\}\text{, if }{a}_1=3,a_n=96\text{ and }{S}_n=189$ then the value of $n$ is

Given, $a_1=3,a_n=96,s_n=189$

Let $r$ be the common ratio of the given GP

$\begin{array}{l}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\therefore \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{a}_n\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=a_1{r}^{n-1}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}96\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=3\cdot r^{n-1}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{r}^{n-1}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{96}{3}=32\end{array}$   (i)

Now, $S_n=\frac{a_1\left(r^n-1\right)}{r-1}$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}189\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{3\left(r^n-1\right)}{r-1}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\left[\because a_1=3\right]\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}63\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{r^n-1}{r-1}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}63(r-1)\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=r^n-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}63r-63\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=r^n-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}63r-63\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=32r-1\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}63r-32r\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=-1+63\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}31r\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=62\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}r\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}=\frac{62}{31}=2\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\end{array}$

From Eqs. (i) and (ii), $n = 6$