(i) Writing $E_n \propto \frac{1}{n^2}$ (ii) Identifying the level to which the electron is emitted. (iii) Calculating the wavelengths and identifying the series of atleast one of the three possible lines, that can be emitted. (i) We have $E_n \propto \frac{1}{n^2}$ (ii) $\therefore$ The energy levels are $-13.6\ \text{eV},-3.4\ \text{eV},-1.5\ \text{eV}$ $\therefore$ The $12.5\ \text{eV}$ electron beam can excite the electron up to $n=3$ level only. (iii) Energy values, of the emitted photons, of the three possible lines are $\;3 \rightarrow 1:(-1.5+13.6)\ \text{eV}=12.1\ \text{eV}$ $\;2 \rightarrow 1:(-3.4+13.6)\ \text{eV}=10.2\ \text{eV}$ $\;3 \rightarrow 2:(-1.5+3.4)\ \text{eV}=1.9\ \text{eV}$ The corresponding wavelengths are : 102 $\text{nm}, 122\ \text{nm}$ and $653\ \text{nm}$ $\left(\lambda=\frac{h c}{E}\right)$