Energy of the electron in the n‌th ‌ state of an atom Here, Z is the atomic number of the atom.
For hydrogen atom, Z=1
Energy required to excite an atom from initial state (ni) to final state (nf),
E‌=−13.6(‌

1

nf2

−‌

1

ni2

)‌eV
⇒‌‌‌

−13.6

nf2

+‌

13.6

ni2

‌=12.9
This energy must be equal to or less than the energy of the incident electron beam.
⇒13.6−12.9‌=‌

13.6

nf2

‌[∵n′=1]
⇒nf‌=4.4‌
State cannot be a fraction number.
⇒‌‌nf=4
Hence, the hydrogen atom would be excited up to 4‌th ‌ energy level.
Rydberg's formula for the spectrum of the hydrogen atom is given by:
‌

1

λ

=R(‌

1

n12

−‌

1

n22

)
Here, λ is the wavelength
Rydberg's canstant, R=1.097×107m−1
For the first member of the Paschen series
‌

1

λ

‌=1.097×107(‌

1

32

−‌

1

42

)
λ‌=18752.4Å
For the first member of Balmer series
n1‌=2,n2=3
‌

1

λ

‌=1.097×107(‌

1

22

−‌

1

32

)
λ‌=6563.3Å