The electron, in a hydrogen atom, initially in a state of quantum number n1
for a hydrogen atom, the energy in terms of principal quantum number 'n' is given as
En = (-13.6/n2) eV
therefore
for n = 1
E1 = -13.6 / 12 = -13.6 eV
and n = 2
E2 = -13.6 / 22 = -3.4 eV
Now,
as per given for state n=2 has the excitation energy of the atom is 10.2 eV. So, the electron is making a transition from n = n1 to n=2 where (n1>2).
Now, we have
En1 – E2 = hc/λ
here
hc/λ = [(6.63 x 10-34 x 3 x 108 ) / (487.5 x 10-3)] x (1/ 1.6 x 10-19) eV
or hc/λ = 2.55 eV
now as
En1= E2 + hc/λ = (-3.4 + 2.55) eV
so,
En1 ≃ - 0.85 eV
But we also have En1 = (-13.6/n12) eV
or
n1 = (-13.6 /En1 )1/2 = (-13.6 / -0.85)1/2
thus, we have
n1 = 4