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Q.
Radiation com in $g$ from transitions $n = 2$ to $n = 1$ of hydrogen atoms fall on $He^+$ ions in $n = 1$ and $n = 2$ states. The possible transition of helium ions as they absorb energy from the radiation is :
Energy released for transition n = 2 to n = 1 of hydrogen atom
$E = 13.6 \; Z^2 ( \frac{1}{n_1^2} - \frac{1}{n^2_2} )$
$Z = 1, n_1 = 1, n_2 = 2$
$E = 13.6 \times 1 \times ( \frac{1}{1^2} - \frac{1}{2^2} ) $
$E = 13.6 \times \frac{3}{4} eV$
For $He^{+} $ ion z = 2
(1) n = 1 to n = 4
$E = 13.6 \times2^{2} \times\left(\frac{1}{1^{2}} - \frac{1}{4^{2}}\right) = 13.6 \times\frac{15}{4} eV $
(2) n = 2 to n = 4
$ E = 13.6 \times2^{2} \times\left(\frac{1}{2^{2}} - \frac{1}{4^{2}}\right) = 13.6 \times\frac{3}{4} eV$
(3) n = 2 to n = 5
$ E = 13.6 \times2^{2} \times\left( \frac{1}{2^{2}} - \frac{1}{5^{2}}\right) = 13.6 \times\frac{21}{25} eV $
(4) n = 2 to n = 3
$ E = 13.6 \times2^{2} \times\left( \frac{1}{2^{2}} - \frac{1}{3^{2}}\right) = 13.6 \times\frac{5}{9} eV $
Energy required for transition of $He^+$ for $n = 2$ to $n = 4$ matches exactly with energy released in transition of $H$ for $n = 2$ to $n = 1$.