In Bohr’s theory the potential energy of an electron at a position is  $\frac{K{r}^2}{2}$ (where  $K$ is a positive constant); then the quantized energy of the electron in nth orbit is: 

Given
Potential energy of an electron  $U=\frac{K{r}^2}{2}$
Where  K=Positive constant
We use force on electron 
$\begin{array}{l}F=\frac{\mathrm{d}U}{\mathrm{d}r}=Kr=\frac{m{v}^2}{r}\left(\because F_c=\frac{m{v}^2}{r}\right)\\ \Rightarrow m{v}^2=K{r}^2\text{ and }mvr=\frac{nh}{2\pi }\\ mr\left(\sqrt{\frac{K}{m}}r\right)=\frac{nh}{2\pi }\Rightarrow r=\sqrt{\frac{nh}{2\pi \sqrt{Km}}}\end{array}$

Energy of n^th orbit is 

$\begin{array}{l}{E}_n=\frac{K{r}^2}{2}+\frac{K{r}^2}{2}=K{r}^2\\ \Rightarrow E_n=\frac{Knh}{2\pi \sqrt{Km}}=\frac{nh}{2\pi }\sqrt{\frac{K}{m}}\end{array}$