Question

Let $a_1,a_2,a_3.........$ are in $G.P.a_r$ denote of $G.P.S_r$ denotes sum to $r$ terms of $G.P.$ If $a_1=3,a_n=96,S_n=189$ then $n$ is

• A. $6$
• B. $4$
• C. $7$
• D. $8$

Answer 1

The correct option is A $6$

We have,

$\begin{array}{c}{a}_1,a_2,a_3........G.P\\ a_1=3a_n=16\\ S_m=189\\ a.{\left(r\right)}^{n-1}=96\\ {\left(r\right)}^{n-1}=32{\left(r\right)}^n=32r\\ \Rightarrow \frac{a_1\left[r^n-1\right]}{r-1}=189\\ \Rightarrow \frac{r^n-1}{r-1}=63\\ \Rightarrow \frac{32r-1}{r-1}=63\\ \Rightarrow 32r-1=63r-63\\ \Rightarrow 62=31r\\ r=2\\ {\left(2\right)}^{n-1}=32\\ n-1=5\\ \therefore n=6\end{array}$

Hence,

Option $A$ is correct answer.