Question

The $3^{rd}$ and $6^{th}$ term of a G.P. is 12 and 96 respectively. If the sum of all terms is 1533, find the number of terms in the G.P.

• A. 8
• B. 10
• C. 9
• D. 12

Answer 1

The correct option is C

9

Given, $a_3=a{r}^2=12\text{and}{a}_6=a{r}^5=96.$

$\Rightarrow \frac{a{r}^5}{a{r}^2}=\frac{96}{12}=8$

$\Rightarrow r^3=8\Rightarrow r=2$

$\therefore a{r}^2=12\Rightarrow a\times 2^2=12$

$\Rightarrow a=3$

Let the number of terms in the GP be $n.$

Sum to $n$ terms of a GP of first term $a$ and common ratio $r(\ne 1)$ is given by
$S_n=\frac{a(r^n-1)}{r-1}.$

$\Rightarrow \frac{3(2^n-1)}{2-1}=1533$

$\Rightarrow 2^n-1=511$

$\Rightarrow 2^n=512$

i.e., $2^n=2^9$

$\therefore n=9$