A bullet loses ((1)/(n)) ^ th of its velocity passing through one...

Let $t$ be the thickness of the plank

${\left\{{υ}_{0}\left(1 - \dfrac{1}{n}\right)\right\}}^{2} = {υ}_{0}^{2} - 2at$

$∴ 2at = {υ}_{0}^{2} - {υ}_{0}^{2}{\left(1 - \dfrac{1}{n}\right)}^{2}$

If $m$ number of planks are required to stop a bullet, then

$0 = {υ}_{0}^{2} - \left(2at\right)m$

$∴ m = {υ}_{0}^{2}2at$

$= {υ}_{0}^{2}{υ}_{0}^{2}\left\{1 - {\left(1 - \dfrac{1}{n}\right)}^{2}\right\}$

$= \dfrac{{n}^{2}}{{n}^{2} - {\left(n - 1\right)}^{2}} = \dfrac{{n}^{2}}{\left(2n - 1\right)}$