Which series of transitions produces the lowest energy photons in the Paschen series?

In the context of the Paschen series, which comprises transitions of electrons to the third energy level (or third excited state) of the hydrogen atom, the energy of the emitted photons is directly related to the difference in energy levels involved in those transitions.

For the lowest energy photons to be produced, the transitions must involve the smallest difference in energy between the upper and lower energy levels. In this series, the transitions that terminate at the third energy level will yield lower energy photons compared to those terminating at higher levels, as the energy difference becomes less significant.

When considering transitions to the second excited state, these involve going from higher energy levels down to n=3 (for example, from n=4 to n=3, or n=5 to n=3). This means that transitions terminating in the second excited state have a smaller energy gap compared to those that start from levels closer to the ground state (such as n=3, n=4, or n=5 to n=2), thereby producing lower energy photons.

In summary, the series of transitions leading to the lowest energy photons in the Paschen series occurs when the transitions end at the second excited state since they involve smaller differences in energy levels, aligning with the principle that