12.5 eV electron beam is used to bombard gaseous hydrogen at room temperature. Also, gaseous hydrogen in its ground state at RT = -13.6 eV

So, the energy of the gaseous hydrogen = -13.6 + 12.5 = -1.1 eV

So, orbital energy E = -13.6/ n² eV  (where, n = 3)

E = -13.6/ 3² = -13.6/9 = -1.5 eV

If the energy is equal to the energy of gaseous hydrogen. It can be concluded that the electron has jumped from n =1 to n - 3 level

In de excitation process, electrons can jump from n=3 to n=1. then Lyman series as

1/ λ = R_y (1/ 1²-1/ n²) , where, R_y =  1.097 x 10⁷ m/s

Wavelength of radiation emitted by transition of electron

For n=3,

We can get

1/ λ = 1.097 x 10⁷ (1/ 1²-1/ 3²) = 1.097 x 10⁷(1- 1/9) = 1.097 x 10⁷x 8/9 = 102.55 nm

If the transmission from n=2 to n=1 , then,

1/ λ = 1.097 x 10⁷ (1/ 1²-1/ 3²) = 1.097 x 10⁷(1- 1/4) = 1.097 x 10⁷x 3/4 = 121.54 nm

If the transmission from n=3 to n=2 , then,

1/ λ = 1.097 x 10⁷ (1/ 2²-1/ 3²) = 1.097 x 10⁷ (1/4- 1/9) = 1.097 x 10⁷x 5/36 = 656.33 nm

Hence, in Lyman series, two wavelengths i.e. 102.55 nm and 121.54 nm are emitted. In Balmer series, one wavelength, 656.33 nm is emitted.