Q.In Bohr's model of the hydrogen atom, the ratio of the maximum frequency to the minimum frequency of light emitted in the Balmer series of the hydrogen spectrum is :

When an individual gaseous element such as hydrogen is excited (by heating or an electrical discharge) and its emitted light is examined through a spectrometer, the result is not a smooth, continuous rainbow of colours but rather a set of sharp, discrete bright lines at very specific wavelengths - a line spectrum, unique to that particular element, functioning almost like a fingerprint that lets scientists identify a gas's composition from its light alone (including the composition of distant stars and the Sun).

For hydrogen specifically, every observed spectral-line wavelength is given by a single Rydberg-type formula,

where $R=1.09737\times10^7\ \text{m}^{-1}$ is the Rydberg constant. Fixing the lower level $n$ and letting $m$ range over every larger integer generates one complete named series of spectral lines: $n=1$ gives the Lyman series (ultraviolet region); $n=2$ gives the Balmer series (visible region - the only series bright enough to be seen directly with the eye); $n=3$ gives the Paschen series (near infrared); $n=4$ gives the Brackett series (mid infrared); and $n=5$ gives the Pfund series (far infrared). Within any one series, successive lines (as $m$ increases) crowd progressively closer together, approaching a limiting wavelength - the series limit - as $m\to\infty$.

This same formula also lets the total number of distinct spectral lines an excited hydrogen sample can emit be counted: if an electron has been excited up to level $n$, it can subsequently make a transition down to any lower level, and the total number of distinct transitions (and hence distinct emitted wavelengths) possible works out to $n(n-1)/2$. …