A monochromatic light is incident on a hydrogen sample in ground state. Hydrogen atoms absorb a fraction of…

Let the higher state be $n$.

The number of wavelengths emitted is given by,

$$\begin{gathered} \frac{n(n-1)}{2}=6 \\ \Rightarrow n^2-n-12=0 \\ \Rightarrow \left(n-4\right)(n+3)=0 \\ \Rightarrow n=4 \end{gathered}$$

For hydrogen atom,
$h\nu =13.6\left(\frac{1}{1}-\frac{1}{16}\right)=\left(13.6\times \frac{15}{16}\right)\mathrm{eV}$$$\begin{gathered} \Rightarrow \nu =\frac{E}{h} \\ =\frac{13.6\times 15\times 1.6\times {10}^{-19}}{16\times 6.62\times {10}^{-34}} \\ =3\times {10}^{15}\mathrm{Hz} \end{gathered}$$