Then from the given condition

\({E_{2n}} - {E_1} = 204\,eV\) or \(\frac{{{E_1}}}{{4{n^2}}} - {E_1} = 204\,eV\)

==> \({E_1}\left( {\frac{1}{{4{n^2}}} - 1} \right) = 204\,eV\)…..\((i)\)

and \({E_{2n}} - {E_n} = 40.8\,eV\)

==> \(\frac{{{E_1}}}{{4{n^2}}} - \frac{{{E_1}}}{{{n^2}}} = {E_1}\left( { - \frac{3}{{4{n^2}}}} \right) = 40.8\,eV\)…..\((ii)\)

From equation \((i)\) and \((ii), \)

\(\frac{{1 - \frac{1}{{4{n^2}}}}}{{\frac{3}{{4{n^2}}}}} = 5\) ==> \(n = 2\)