The ionization energy of the electron in the hydrogen atom in its ground state is $13.6\text{eV}$. The atoms are excited to higher energy levels to emit radiations of $6$ wavelengths. Maximum wavelength of emitted radiation corresponds to the transition between:

1. Determine the level $n$:The number of spectral lines emitted is given by $\frac{n(n-1)}{2}=6$.$n^2-n-12=0⟹(n-4)(n+3)=0⟹n=4$.So, the atoms are in the $4^{th}$ energy level.2. Determine maximum wavelength:Maximum wavelength corresponds to minimum energy change ($\Delta E$).Possible transitions are $4\to 3,4\to 2,4\to 1,3\to 2,3\to 1,2\to 1$.The smallest energy gap among these is the one with the highest principle quantum numbers, i.e., $n=4$ to $n=3$.