As the electron revalues in an stationary orbit,
Thus centrifugal force acting on the electron to balanced by the coulombic force
Fcentrifugal = Fcoulomb
\(\frac{mv^2}{r_n}=\frac{kl^2}{{r_n}^2}\)
\(r_n=\frac{kl^2}{m{v_n}^2}\)
We take
\(m\,v_nr_n= \frac{nh}{2\pi}\)
\(\because V_n = \frac{nh}{2\pi mr_n}\)
Then,
\(r_n = \frac{kl^2}{m{v_n}^2}\)
\(r_n = \frac{kl^2}{m}.\frac{(2\pi mr_n)^2}{n^2h^2}\)
\(r_n = \frac{kl^2 4\pi^2m^2{r_n}^2}{m\,n^2h^2}\)
\(r_n = \frac{n^2h^2}{kl^24\pi^2m}\)
\(r_n \propto n^2\)