Let $t$ be the thickness of the plank

${\left\{{\upsilon }_0\left(1-\frac{1}{n}\right)\right\}}^2={\upsilon }_0^2-2at$

$\therefore 2at={\upsilon }_0^2-{\upsilon }_0^2{\left(1-\frac{1}{n}\right)}^2$

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

$0={\upsilon }_0^2-\left(2at\right)m$

$\therefore m=\frac{{\upsilon }_0^2}{2at}$

$=\frac{{\upsilon }_0^2}{{\upsilon }_0^2\left\{1-{\left(1-\frac{1}{n}\right)}^2\right\}}$

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