Which energy-level change shown in the diagram below emits electromagnetic radiation with the longest wavelength?

a) an electron moving from 4 to 5
b) an electron moving from 5 to 2
c) an electron moving from 6 to 1
d) an electron moving from 2 to 1

-- Electromagnetic radiation is emitted (spit out) when an electron
moves from a higher energy level to a lower one.

-- The greater the difference in energy between the two levels, the more
energy is emitted, and the shorter the wave;length of the emitted photon.

a) an electron moving from 4 to 5
This electron is moving to a HIGHER energy level.  It doesn't emit anything.
In fact, you had to shoot energy INTO the atom to make the electron do this.

b) an electron moving from 5 to 2
This electron loses maybe something like  3.1 eV.

c) an electron moving from 6 to 1
This electron loses maybe something like  13.4 eV.

d) an electron moving from 2 to 1
This electron loses something like  10.2 eV.

The least energy is emitted by the electron in choice-b.
That one will emit the photon with the longest wavelength (shortest frequency).

This whole chart refers to the one electron in a hydrogen atom ...
how much energy it has at each level, how much energy is required
to boost it from one level to another, and how much energy it spits out
when it falls from one energy level to another.

The Balmer Series ... any drop that ends up at energy-level #2 ...
is the set of transitions that produce photons of visible light.
 
The Paschen Series are transitions that all emit less energy.
All of those emit infrared photons.

The Lyman series are transitions that all emit big energy.
All of those emit X-ray photons.

Did you ever figue this