The hydrogen spectrum is a series of lines that are produced when the electrons in hydrogen atoms transition between different energy levels. These transitions result in the emission or absorption of light at specific wavelengths.

The ultraviolet region of the hydrogen spectrum corresponds to transitions where the electrons move from higher energy levels to lower energy levels. The largest wavelength in this region corresponds to the smallest energy difference between the two levels involved in the transition. In other words, it represents the transition with the lowest energy.

The formula to calculate the wavelength of a transition in the hydrogen spectrum is given by the Rydberg formula:

1/λ = R_H * (1/n1^2 - 1/n2^2)

Where λ is the wavelength, R_H is the Rydberg constant for hydrogen (approximately 1.097 * 10^7 m^-1), and n1 and n2 are the principal quantum numbers of the initial and final energy levels, respectively.

To find the largest wavelength in the ultraviolet region, we need to find the transition with the lowest energy. This corresponds to the transition with the smallest difference in the values of n1 and n2. Since the smallest possible value for n2 is 1, we need to find the largest possible value for n1.

The largest possible value for n1 can be determined by using the fact that the energy of an electron in a hydrogen atom is given by the equation:

E = -13.6 eV / n^2

Where E is the energy and n is the principal quantum number. Setting E equal to zero (since we are looking for the highest energy level), we can solve for n:

0 = -13.6 eV / n^2
n^2 = 13.6 eV / 0
n^2 = ∞
n = ∞

Therefore, the largest possible value for n1 is infinity.

Plugging in the values into the Rydberg formula, we get:

1/λ = 1.097 * 10^7 m^-1 * (1/infinity^2 - 1/1^2)
1/λ = 1.097 * 10^7 m^-1 * (0 - 1)
1/λ = -1.097 * 10^7 m^-1
λ = -1/(-1.097 * 10^7 m^-1)
λ = 9.11 * 10^-8 m
λ = 91.1 nm

Therefore, the largest wavelength in the ultraviolet region of the hydrogen spectrum is 91.1 nm, which is less than the given value of 122 nm.

Moving on to the infrared region of the hydrogen spectrum, this corresponds to transitions where the electrons move from lower energy levels to higher energy levels. The smallest wavelength in this region corresponds to the transition with the highest energy difference between the two levels involved in the transition. In other words, it represents the transition with the highest energy.

Using the Rydberg formula, we can calculate the smallest wavelength in the infrared region by finding the transition with the highest energy difference. This corresponds to the transition with the largest difference in the values of n1 and n2. Since the smallest possible value for n1 is 1, we need to find the largest possible value for n2.

Again, the largest possible value for n2 can be determined by using the fact that the energy of an electron in