Light shines through atomic hydrogen gas. It is seen that the gas absorbs light readily at a wavelength of 91.63 nm. What is the value of n at the...

The Rydberg equation should be used to solve this problem:

where lambda is the wavelength, R is the Rydberg constant, n-subf is the final n-level, and n-subi is the initial n-level.

We know that lambda = 91.63nm, R = 1.097 x 10e7, and n-subi = 1.

 So, 

1/1.005 = .995

1/n_fe2 = .005

1/.005 = n_fe2

sqrt (200) = n_f

The square root of 200 is closer to 14.14, so there's some rounding in the final answer, since n must be an integer. This error is mostly due to the inconsistent significant figures and abbreviated forms of the values for R and lambda.