**The Lyman and Balmer series**

The Lyman and Balmer series are two important series of spectral lines in the hydrogen atom and hydrogen-like ions. These series are a result of transitions between different energy levels in the atom or ion.

The Lyman series consists of spectral lines corresponding to transitions where the electron moves from higher energy levels to the ground state (n=1). The Balmer series, on the other hand, corresponds to transitions where the electron moves from higher energy levels to the first excited state (n=2).

**Wavelength of the first line of Lyman series**

The wavelength of the first line of the Lyman series (Lyman alpha) is given by the Rydberg formula:

1/λ = R (1 - 1/n²)

where λ is the wavelength, R is the Rydberg constant, and n is the principal quantum number of the final energy level.

For the Lyman alpha transition, the final energy level would be n=1 (ground state), and substituting these values into the formula, we can find the wavelength.

**Wavelength of the second line of Balmer series**

The wavelength of the second line of the Balmer series (Balmer beta) is also given by the Rydberg formula:

1/λ = R (1 - 1/n²)

However, in this case, the final energy level would be n=3 (first excited state). Substituting these values into the formula, we can find the wavelength.

**Equating the wavelengths**

Given that the wavelength of the first line of the Lyman series is identical to that of the second line of the Balmer series for the same hydrogen-like ion X, we can equate the two wavelengths:

1/λ_lyman = R (1 - 1/1²)
1/λ_balmer = R (1 - 1/3²)

Since the wavelengths are identical, we can set the two equations equal to each other:

R (1 - 1/1²) = R (1 - 1/3²)

Simplifying the equation, we get:

(1 - 1/1²) = (1 - 1/3²)

1 - 1 = 1 - 1/9

0 = 8/9

This equation is not possible, as it leads to a contradiction. Therefore, there must be an error in the question.