Show that the first few frequencies of light that are emitted when electrons falls to the nth level form levels higher than n, are approximate harmonics (i.e., in the ratio 1 : 2: 3...) when `n gt gt 1`.

Explanation

According to Bohr.s formula , the frequency of electromagnetic radiation emitted when electron falls from `p^(th)` level to the `n^(th)` level (p > n) is given by
`v_"pn"=cRZ^2[1/(n+k)^2-1/n^2]`
where p=n+k and k=1,2,3,….
R is the Rydberg.s constant
Z is the atomic number of atom
As k < < n,
`v_"pn"=cRZ^2[1/n^2(1+k/n)^(-2)-1/n^2]`
`=cRZ^2[1/n^2(1-(2k)/n)-1/n^2]`
`=cRZ^2[1/n^2 -(2k)/n^3-1/n^2]`
`=cRZ^2(2k)/n^3=((2cRZ^2)/n^3)k`
Since k=1,2,3,..., therefore , `v_"pn"` are approximate harmonics in the ratio 1:2:3....