Explanation:
The electron moves around the nucleus due to centripetal force provided by the electrostatic force, which is given by
\({F_e} = \frac{{Z{e^2}}}{{4\pi {\varepsilon _0}r_n^2}} \Rightarrow {F_e} \propto \frac{1}{{r_n^2}}\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\)
Also, radius of orbit,
\({r_n} = \left( {\frac{{{h^2}{\varepsilon _0}}}{{\pi m{e^2}}}} \right)\frac{{{n^2}}}{Z}\)
\( \Rightarrow {r_n} \propto {n^2}\,\,\,\,\,\,\,\,\,\,\,\,(2)\)
From eq.(1) and eq.(2), we get
\({F_e} \propto \frac{1}{{{{\left( {{n^2}} \right)}^2}}} \Rightarrow {F_e} \propto {n^{ - 4}}\)