Five possible transitions for a hydrogen atom are listed below: Select whether the atom gains or loses energy for each transition.
a.
Loses Gains ni = 6; nf = 4
Loses Gains ni = 4; nf = 2
Loses Gains ni = 5; nf = 3
Loses Gains ni = 2; nf = 5
Loses Gains ni = 5; nf = 7
b. Find the transition where the atom gains the most energy. How much energy does the atom gain?
c. Find the transition which will emit the shortest wavelength photon. What is the energy of the photon of this transition

 

You have not explained your notation which is by no means standard. I'm guessing that ni is the principal quantum number. I can't tell from your post what nf refers to. It is either the orbital or the angular momentum.