How many different wavelength may be observed in the spectrum from a hydrogen sample if the atoms excited to states with principal quantum number n?

Explanation

Suppose the electron is in its `n^(th)` state then it can make transition to `(n-1)^(th)` state, `(n-2)^(th)` state,.., `2^(nd)` state or `1^(st)` state. Hence, there are total (n-1) possible transitions. Similarly for electron in `(n-1)^(th)` state possible transitions are `(n - 2)^(th)` and for `(n-2)^(th)` state there are (n-3) possible transitions and so on. Hence total possible number of transitions can be calculated as follows:
Total transitions (n - 1)+ (n - 2) + (n -3) +.. +3+2+1
Hence, n(n - 1)/2 number of different possible wavelengths can be emitted from the sample of hydrogen.