A monochromatic light is incident on a hydrogen sample in ground...

Let the higher state be $n$.

The number of wavelengths emitted is given by,

$\dfrac{n ( n - 1 )}{2} = 6 ⇒ {n}^{2} - n - 12 = 0 ⇒ \left(n - 4\right) ( n + 3 ) = 0 ⇒ n = 4$

For hydrogen atom,
$hν = 13 . 6\left(\dfrac{1}{1} - \dfrac{1}{16}\right) = \left(13 . 6 \times \dfrac{15}{16}\right)eV$$⇒ ν = \dfrac{E}{h} = \dfrac{13 .6 \times 15 \times 1 . 6 \times {10}^{- 19}}{16 \times 6 . 62 \times {10}^{- 34}} = 3 \times {10}^{15}Hz$