Question

If in a G.P. $\left\{a_n\right\},a_1=3,a_n=96$ 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 $6$

For a G.P $a_n=a_1{r}^{n-1}$ where r is common ratio.

Given that $a_n=96$

$\Rightarrow 3\times r^{n-1}=96$

$\Rightarrow r^{n-1}=32$

$\Rightarrow r^n=32r$ [Let this be equation 1]

Also, sum of the first n terms of a finite G.P. is given by $S_n=\frac{a_1(r^n-1)}{r-1}$

$\Rightarrow \frac{3(32r-1)}{r-1}=189$

$\Rightarrow 32r-1=63r-63$

$\Rightarrow 31r=62$

$\Rightarrow r=2$

Putting $r=2$ in equation 1,we get

$\Rightarrow 2^n=64$

$\Rightarrow n=6$

Hence the answer is option $B$