Consider the follwoing two elerctronic transition possibilites in a hydrogen atom as pictured below :

(1) The electron drops from third Bohr's orbit to second Bohr's obit followed with the next transition from second to first Bohr's orbit .
(2) The electron drops from third Bohr's orbit to first Bohr's orbit directly .
Show that :
(a) The sum of the enrgies for the transitions ` n=3` to ` n=2` and ` n=2` to ` n=1` is equal to the energy of transiton for ` n=3` to ` n=1` .
(b) Are wavelengths and frequencies of the emitted spectrum also additive in the same way as their energies are ?

Explanation

Applying, `DeltaE=R_(H)[(1)/(n_(1)^(2))-(1)/(n_(2)^(2))]`
For n=3 to n=2,
`DeltaE_(3to2)=R_(H)[(1)/(2^(2))-(1)/(3^(2))]=R_(H)xx(5)/(36)` . . .(i)
For n=2 to n=1,
`DeltaE_(2to1)=R_(H)[(1)/(1^(2))-(1)/(2^(2))]=R_(H)xx(3)/(4)` . . .. (ii)
For n=3 to n=1,
`DeltaE_(3to1)=R_(H)[(1)/(1^(2))-(1)/(3^(2))]=R_(H)xx(8)/(9)` . . .(iii)
Adding equations (i) and (ii).
`R_(H)((5)/(36)+(3)/(4))=R_(H)((5+27)/(36))=R_(H)xx(8)/(9)`
thus, `DeltaE_(3to1)=DeltaE_(3to2)+DeltaE_(2to1)`.