શ્રેણી $\quad 2,2 \sqrt{2}, 4, \ldots$ નું કેટલામું પદ $128$ થાય ?

The given sequence is $2,2 \sqrt{2}, 4 \ldots \ldots$ is $128 ?$

Here, $a=2$ and $r=(2 \sqrt{2}) / 2=\sqrt{2}$

Let the $n^{\text {th }}$ term of the given sequence be $128 .$ 

$a_{n}=a r^{n-1}$

$\Rightarrow(2)(\sqrt{2})^{n-1}=128$

$\Rightarrow(2)(2)^{\frac{n-1}{2}}=(2)^{7}$

$\Rightarrow(2)^{\frac{n-1}{2}+1}=(2)^{7}$

$\therefore \frac{n-1}{2}+1=7$

$\Rightarrow \frac{n-1}{2}=6$

$\Rightarrow n-1=12$

$\Rightarrow n=13$

Thus, the $13^{\text {th }}$ term of the given sequence is $128$