Note: I'm assuming that this is a hydrogen electron transition.

5.13 μm = 5.13 x 10⁻⁶ m

Rydberg wavelength equation:

1/λ = R * [(1/nf²) - (1/ni²)]

R = Rydberg constant in meters = 1.097 x 10⁷ m⁻¹

ni = initial energy level

nf = final energy level

In this case, nf = 6, λ = 5.13 x 10⁻⁶ m .

Solve for ni:

1 / (5.13 x 10⁻⁶ m) = (1.097 x 10⁷ m⁻¹) * [(1/6²) - (1/ni²)]

1.949 x 10⁵ m⁻¹ = (1.097 x 10⁷ m⁻¹) * [(1/6²) - (1/ni²)]

0.01777 = (1/36) - (1/ni²)

-0.01001 = -1/ni²

ni² = 1 / 0.01001

ni² = 99.90

ni = 9.995

Initial energy level was n = 10 .