Show that the first few frequencies of light that is emitted when electrons fall to the `n^(th)` level from levels higher than n, are approximate harmonics (ie, in the ratio 1:2:3...) when `n gt gt 1`.

Explanation

In an atom of an element with atomic no. 2, when an electron makes a transition from `(n + x)^(th)` orbit (where x = 1, 2, 3, ....) to `n^(th)` orbit, if frequency and wavelength of emitted radiation are respectively f and then,
`(1)/(lambda)=RZ^(2)((1)/(n^(2))-(1)/((n+x)^(2)))`
`:.(f)/(c)=RZ^(2){((n+x)^(2)-n^(2))/((n+x)^(2)(n)^(2))} ( :. c=flambda)`
`:.f=RcZ^(2){(n^(2)+2nx+x^(2)-n^(2))/((n+x)^(2)(n^(2)))}`
`:.f=RcZ^(2){(2nx+x^(2))/((n+x)^(2)(n^(2)))}`
Here `n gt gt 1 and x = 1,2,3.... and so (n+x)= n( :. x lt lt n)`. Also we can neglecte `x^(2)` from the addition. Thus,
`r~~RcZ^(2)((2nx)/(n^(4)))`
`:.f~~((2rCZ^(2))/(n^(3)))x`
`:.f prop x`
`:.f_(1):f_(2):f_(3)=x_(1):x_(2):x_(3)`
`=1:2:3`