Every time hydrogen emits a photon in the Balmer series, it is...

The Balmer series corresponds to the transition of an electron from a higher energy level ($n \ge 3$) to the first excited state ($n = 2$).

After this transition, the electron is in the $n = 2$ state. It will subsequently transition to the ground state ($n = 1$), emitting a photon in the Lyman series.

The energy difference for the transition from $n = 2$ to $n = 1$ is given by:

$\Delta E = 13.6 \left( \dfrac{1}{1^2} - \dfrac{1}{2^2} \right) \text{ eV} = 13.6 \times \dfrac{3}{4} \text{ eV} = 10.2 \text{ eV}$

The wavelength of the emitted photon is:

$\lambda = \dfrac{hc}{\Delta E} = \dfrac{1242 \text{ eV nm}}{10.2 \text{ eV}} \approx 121.7 \text{ nm}$

Rounding to the nearest integer, we get $122 \text{ nm}$.