Show that the first few frequencies of light that are emitted when electrons fall 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

When on electron jumps form mth orbit to nth orbit, the frequency of electromagnetic radiation emitted,
according to Bohr's formula, is `v =cRZ^2 [1/(m^2)-1/(n^2)]` where R is Raydberge constant.
Let us take `m=(m+p), "where p"=1,2,3....`
`v =cRZ^2 [1/(n+p)^2-1/(n^2)]=cRZ^2 [1/(n^2)(1+p/n)^(-2)-1/(n^2)]`
As `P lt lt n` therefore, `v =cRZ^2 [1/(n^2)(1-(2p)/n)-1/(n^2)]=-cRZ^2(2p)/(n^3)=((-2cRZ^2)/(n^3))p`
As `P=1,2,3.....`, therefore, first few frequency of light emitted are approx. harmonics (i.e., in the ratio 1:2:3.....), when `ngt gtp`.