An electron in the n = 5 level of an h atom emits a photon of wavelength 1282.17 nm. to what energy level does the electron move?

This is an interesting (read tricky!) variation of Rydberg Eqn calculation.

Rydberg Eqn: 1/λ = R [1/n1^2 - 1/n2^2]

Where λ is the wavelength of the light; 1282.17 nm = 1282.17*10^-9 m

R is the Rydberg constant: R = 1.09737*10^7 m-1

n2 = 5 (emission)

Hence 1 / (1282.17 * 10^-9) = 1.09737 * 10^7 [1/n1^2 - 1/25^2]

Some rearranging and collecting up terms:

1 = (1282.17 * 10^-9) (1.09737 * 10^7) [1/n2 - 1/25]

1 = 14.07[1/n^2 - 1/25]

1 = 14.07/n^2 - (14.07/25)

14.07n^2 = 1 + 0.5628

n = √ (14.07/1.5628) = 3