This problem involves the Hydrogen Spectrum and Bohr's Model, covered in Unit 2: Atomic Structure (Chemistry) and Unit 18 (Physics). The sources confirm the energy levels and spectral series of hydrogen .

1. Determine the excited state ($n$): The number of spectral lines emitted when an electron drops from the $n$-th state to the ground state is given by the formula $\frac{n(n-1)}{2}$. Given 6 wavelengths, $\frac{n(n-1)}{2} = 6 \Rightarrow n^2 - n - 12 = 0 \Rightarrow (n-4)(n+3) = 0$. Thus, the electron is excited to the $n=4$ state.
2. Relate Wavelength to Energy: The wavelength ($\lambda$) of emitted radiation is inversely proportional to the energy difference of the transition ($\Delta E$), since $\Delta E = h\nu = \frac{hc}{\lambda}$ . To find the maximum wavelength, we must identify the transition with the minimum energy difference.
3. Identify the Transition: Energy levels in the hydrogen atom become closer together as $n$ increases ($E_n \propto -1/n^2$) . Therefore, the smallest energy gap involving the excited state $n=4$ is the transition to the immediate lower level, i.e., from $n=4$ to $n=3$. This corresponds to the first line of the Brackett series (infrared region) .