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

Explanation

From the nth , the atom may go to `(n = 1)th` state , … second state or first state , So there are `(n - 1)` possible transition starting from the nth state. The atom reaching `(n - 1)th` state may make `(n = 2)` differente transition and so on. In general, the total number of possible transition is
`(n - 1) + (n - 2) + (n - 3) + .....+ 2 + 1 = (n(n - 1))/(2)`