How does the separation between the energy states vary on increasing the principle quantum number (n) in a hydrogen atom?

Decreases

The energy of the hydrogen atom is given by 

E_n = -13.6/n² 

E₂-E₁ = 10.2 eV

E₃-E₂ = 1.89 eV

E₄-E₃ = 0.66 eV

So the enegy level diagram can shown as

So, the energy difference between the states decreases with increasing n.