How many terms of the G.P. 3, 3/2, 3/4, ... be taken together to make

Here, a = 3

Common ratio, $r=\frac{1}{2}$

$S_{n}=\frac{3069}{512}$

$\therefore S_{n}=3\left\{\frac{1-\left(\frac{1}{2}\right)^{n}}{1-\frac{1}{2}}\right\}$

$\Rightarrow \frac{3069}{512}=3\left\{\frac{1-\frac{1}{2^{n}}}{\frac{1}{2}}\right\}$

$\Rightarrow \frac{3069}{512}=6\left\{1-\frac{1}{2^{n}}\right\}$

$\Rightarrow \frac{3069}{3072}=1-\frac{1}{2^{n}}$

$\Rightarrow \frac{1}{2^{n}}=1-\frac{3069}{3072}$

$\Rightarrow \frac{1}{2^{n}}=\frac{3}{3072}$

$\Rightarrow 2^{n}=\frac{3072}{3}$

$\Rightarrow 2^{n}=1024$

$\Rightarrow 2^{n}=2^{10}$

$\therefore n=10$