Answer:
The Balmer series is given by
1 / λ = R (1 / 2^2 - 1 / n^2)
where R is the Rydberg constant of 1.0968E-3 / Angstroms
(1 Angstrom = 1.0E-1 nm = 10E-10m)
Let x = (1 / 4 - 1 / n^2)
x = 1 / (R λ)
Note 410 nm = 4100 A
x = 1 / (1.0968E-3 * 4.1E3) = 1 / (1.0968 * 4.1) = .222
If we write
x = 1/4 - 1/n^2 = .222
1/n^2 = 1/4 - .222 = .0276
n^2 = 36.2
So n = 6
(Since modern terminology gives wavelengths in nanometers - it might be helpful to write R = 1.0968E7 m-1)
