**The Frequency of the First Line of Balmer Series in Hydrogen Atom and its Corresponding Line in Singly Ionized Helium Atom**

The Balmer series is a series of spectral lines in the visible region of the electromagnetic spectrum that are emitted by hydrogen atoms when electrons transition between higher energy levels to the second energy level (n=2). The frequency of the first line of the Balmer series in a hydrogen atom is denoted as v0.

**Frequency of the First Line of Balmer Series in Hydrogen Atom (v0)**

The frequency of the first line of the Balmer series in a hydrogen atom can be calculated using the Balmer series formula:

1/λ = R_H * (1/2^2 - 1/n^2)

Where λ is the wavelength, R_H is the Rydberg constant for hydrogen (approximately 1.097 x 10^7 m^-1), and n is the energy level the electron transitions from. In this case, n=3 since the electron transitions from the third energy level (n=3) to the second energy level (n=2).

By substituting the values into the formula, we can calculate the wavelength (λ) of the first line of the Balmer series in hydrogen. The frequency (v) can be determined using the equation v = c/λ, where c is the speed of light (approximately 3 x 10^8 m/s).

**Frequency of Corresponding Line in Singly Ionized Helium Atom**

Singly ionized helium (He+) is an atom that has lost one of its electrons. The Balmer series formula is applicable not only to hydrogen but also to singly ionized helium since both atoms have a similar electron configuration.

To find the frequency of the corresponding line emitted by singly ionized helium, we use the same formula as for hydrogen, but with the appropriate values for helium. The difference lies in the value of the Rydberg constant, which is different for helium. The Rydberg constant for singly ionized helium (R_He+) is approximately 4.56 x 10^6 m^-1.

By substituting the values into the Balmer series formula, we can calculate the wavelength (λ) of the corresponding line in singly ionized helium. The frequency (v) can then be determined using the equation v = c/λ, where c is the speed of light.

In conclusion, the frequency of the first line of the Balmer series in a hydrogen atom is v0. The frequency of the corresponding line emitted by singly ionized helium can be calculated using the same Balmer series formula with the appropriate values for helium, including the Rydberg constant for singly ionized helium.