Consider the following two possibilities for electron tran$-$

sitions in a hydrogen atom, pictured below: $(1)$ The electron

drops from the Bohr orbit $n=3$ to the orbit $n=2,$ followed by

the transition from $n=2$ to $n=1.(2)$ The electron drops from

the Bohr orbit $n=3$ directly to the orbit $n=1.$

$($a$)$ Show that the sum of the energies for the transitions $n=$

$3n=2$ and $n=2n=1$ is equal to the energy of

the transition $n=3n=1.$

$($b$)$ Are either the wavelengths or the frequencies of the emit$-$

ted photons additive in the same way as their energies?

Explain.