(1) Show that the sum of energies for the transition from n=3 to n=2 and from n=2 to n=1 is equal to the energy of transition from n=3 to n=1 in case of an H-atom. (2) Are wavelength and frequencies of the emitted spectrum also additive as their energies?

Explanation

(1) `DeltaE_(3to2)=R(_H)((1)/(2^(2))-(1)/(3^(2)))` . . . . [1]
`DeltaE_(2to1)=R_(H)((1)/(1^(2))-(1)/(2^(2)))` . . . [2]
`DeltaE_(3to1)=R_(H)((1)/(1^(2))-(1)/(3^(2)))` . . . [3]
From equation (1), (2) and (3) we have,
`DeltaE_(3to2)+DeltaE_(2to1)=R_(H)((1)/(1^(2))-(1)/(3^(2)))=DeltaE_(3to1)`
Thus, `DeltaE_(3to1)=DeltaE_(3to2)+DeltaE_(2to1)`
(2) Since, E=hv hence frequencies are also additive
but `E=(hc)/(lamda)` and thus wavelengths are not additive.