n=3 —> n=1
n=4 —> n=2
n=7 —> n=5
n=10 —> n=8
Can you explain how you get the answer? Please and thank you in advance
2 Answers

n=3 —> n=1
The explanation requires a number of steps…
The highest frequency means the biggest photon energy (E=hf).
All the junps in the list of choices involve dropping down 2 levels.
The biggest photon energy means then biggest gap (energy difference) between the 2 energy levels.
The gap between energy levels gets smaller as n gets bigger – see diagram in link.
So the biggest gap is between the smallest n values.
The smallest n values are n=3 —> n=1.
Look at the diagram carefully and it should be clear.

Photon energy = energy of the difference between two energy levels. But you also know the photon energy E is
E = hf where h = planck’s constant and f = frequency.
Now the energy difference between two states is DE = E0*(1/n1^2 – 1/n2^2) where E0 = ground state
So the photon energy is DE and the photon frequency is
f = (E0/h)(1/n1^2 – 1/n2^2) –> so all that matters is how big the number 1/n1^2 – 1/n2^2 is. So let’s make a table
n1 n2 (1/n1^2 – 1/n2^2)
3 1 8/9
4 2 3/16
7 5 24/1225
10 8 9/1600
Clear the transition from n=3 to n=1 has an energy closest to the ground state so that is the transition whose photon has the highest frequency