In what principle quantum number did the electron originally reside?

The energy h.nu of the radiated light is given by

h.nu = 6.626 x 10^-34 x 1.141 x 10^14 = 2.178 x 10^-18.(1/4^2 - 1/n^2) J

or 1/4^2 - 1/ n^2 = 3.4712 x 10^-2 which gives n = 5.999

so the light originated from the energy level with principal quantum number n = 6.

Note added: My apologies, I could have been more explicit about the logic behind this calculation. We are told that the energy in Joules for an energy level of principal quantum number n in hydrogen atoms is given by

E(n) = -2.178 x 10^-18 (Z^2/n^2)

Note that zero energy corresponds to a free electron, one detached from the atom, hence the minus sign for bound electrons. The photon of energy h.nu is produced by a transition between two energy levels, with the lower having n = 4, and we need to find n for the upper from

h.nu = E(4) - E(n) = 2.178 x 10^-18 (1/4^2 - 1/n^2)

as Z = 1 for hydrogen atoms. The photon energy, from Planck's constant h and the frequency nu of the light emitted, divided by 2.178 x 10^-18 gives 3.4712 x 10^-2, which enables n to be derived from the last equation in the text above.

Principal Quantum Level