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

The frequency of any line in a series in the spectrum of hydrogen like atoms corresponding to the transition of electrons from `(n+p)` level to nth level can be expressed as a difference of two terms,
`v_(min) = cRZ^(2)[(1)/((n+p)^(2)) - (1)/(n^(2))]`
where, `m = n + p, (p = 1, 2, 3"....")`, and R is Ryberg constant.
For `p lt lt n`
`v_(mn) = cRZ^(2) [(1)/(n^(2))(1+(p)/(n))^(-2) - (1)/(n^(2))]`
`v_(mn) = cRZ^(2) [(1)/(n^(2)) - (2p)/(n^(3)) - (1)/(n^(2))]`
[By binomial theorem `(1+x)^(2) = 1 + nx "if" |x| lt 1`]
`v_(mn) = cRZ^(2) (2p)/(n^(3)) = ((2cRZ^(2))/(n^(3)))p`
Thus, the first few frequencies of light that is emitted when electrons fall to the nth level from levels higher than n, are approximate harmonic (i.e., `1 : 2 : 3 "....."`) when `n gt gt 1`