Showing that the radius of orbit varies as ${\mathit{n}}^2$
$\frac{\mathit{m}{\mathit{v}}^2}{\mathit{r}}=\frac{1}{4\mathit{\pi }{\mathit{\epsilon }}_0}\frac{{\mathit{e}}^2}{{\mathit{r}}^2}$
or $\mathit{m}{\mathit{v}}^2\mathit{r}=\frac{1}{4\mathit{\pi }{\mathit{\epsilon }}_0}{\mathit{e}}^2$ .......(i)
$\mathit{m}\mathit{v}\mathit{r}=\frac{\mathit{n}\mathit{h}}{2\mathit{\pi }}$
${\mathit{m}}^2{\mathit{v}}^2{\mathit{r}}^2=\frac{{\mathit{n}}^2{\mathit{h}}^2}{4{\mathit{\pi }}^2}$ .......(ii)
Divide (ii) by (i)
$\mathit{m}\mathit{r}=\frac{{\mathit{n}}^2{\mathit{h}}^2}{4{\mathit{\pi }}^2}\times \frac{4\mathit{\pi }{\mathit{\epsilon }}_0}{{\mathit{e}}^2}$
$\therefore \mathit{r}=\frac{{\mathit{n}}^2{\mathit{h}}^2}{4{\mathit{\pi }}^2\mathit{m}{\mathit{e}}^2}\cdot 4\mathit{\pi }{\mathit{\epsilon }}_0$
$\therefore \mathit{r}\propto {\mathit{n}}^2$
(Give full credit to any other correct alternative method)