Consider your observations for Li and Na, the first two alkali metals: each showed only a single emission line. The ground-state electron configuration for these metals is [He]2s and [Ne]3s, respectively. Since there is only one electron in their valence shells (the rest of the electrons are core electrons), they are "almost" hydrogenic atoms, and we can try and use the Rydberg formula to explain their spectra. However, the emission line you observed for each does not correspond to a transition involving a change in principal quantum number, n. Rather, each involves a change in magnetic quantum number, l. We need to make some adjustments to get this to work.
1) I've provided the ground-state electronic configurations of these elements. Please write the excited-state electron configurationfrom which these elements emitted photons. (You may find this link useful, if you haven't already found something similar: https://physics.nist.gov/PhysRefData/ASD/lines_form.html Note that the spectroscopic notation here indicates a neutral atom as, e.g., Na i, a singly-ionized atom as Na ii, a doubly-ionized atom as Na iii, etc. It's just a historical convention.)