The wave numbers of three spectral lines of $H$ atom are considered. Identify the set of spectral lines belonging to the Balmer series. ($R =$ Rydberg constant)

The frequency of a certain line of the Lyman series of the atomic spectrum of the $H$ atom satisfies the following conditions:
$(i)$ It is the sum of the frequency of another Lyman line and a Balmer line.
$(ii)$ It is the sum of the frequency of a certain line,a Lyman line,and a Paschen line.
$(iii)$ It is the sum of the frequency of a Lyman and a Paschen line but no Brackett line.
To what transition does this frequency correspond?

The energy of an electron in the second Bohr orbit of a hydrogen atom is:

If the longest wavelength of the spectral line of the Paschen series of $Li^{2+}$ ion spectrum is $x \ \mathring{A}$,then the longest wavelength (in $\mathring{A}$) of the Lyman series of the hydrogen spectrum is

$(i)$ The energy associated with the first orbit in the hydrogen atom is $-2.18 \times 10^{-18} \, J \, atom^{-1}$. What is the energy associated with the fifth orbit?
$(ii)$ Calculate the radius of Bohr's fifth orbit for the hydrogen atom.

In astronomical observations,signals observed from the distant stars are generally weak. If the photon detector receives a total of $3.15 \times 10^{-18} \, J$ from the radiations of $600 \, nm$,calculate the number of photons received by the detector.