Question

If in a geometric progression $a_n,a_1=3,a_n=96$ and$S_n=189$ then the value of $n$ is

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

The correct option is B

$6$

Explanation for correct option:

Find the value of $n$:

Given,

$$\begin{gathered} a_1=3 \\ {a}_n=96 \\ {S}_n=189 \end{gathered}$$

The$n^{th}$ term of GP is $a{r}^{n-1}$

$$\begin{gathered} 3\left(r^{n-1}\right)=96 \\ \Rightarrow r^{n-1}=\frac{96}{3} \\ \Rightarrow r^{n-1}=32 \\ \Rightarrow r^n=32r\dots \left(i\right) \end{gathered}$$

Now,

$$\begin{gathered} S_n=\frac{a(r^n-1)}{(r-1)} \\ \Rightarrow \frac{a(r^n-1)}{(r-1)}=189 \\ \Rightarrow \frac{3(32r-1)}{(r-1)}=189 \\ \Rightarrow \frac{(32r-1)}{(r-1)}=63 \\ \Rightarrow 32r-1=63r-63 \\ \Rightarrow 31r=62 \\ \Rightarrow r=2 \end{gathered}$$

Substituting the value of $r$ in equation $\left(i\right)$, we get

$$\begin{gathered} 2^n=64 \\ \Rightarrow 2^n=2^6 \\ \Rightarrow n=6 \end{gathered}$$

Hence, the correct option is B.