Number of spectral lines from $n_2$ to $n_1$ [closed]
Solution 1:
When electron excited to n state. It can be dexcited to $n-1, n-2. n-3 , ......... 2,1.$ So total number of spectral lines in this case is$$ 1+2+3+......+(n-1)+ n = \frac {n(n+1)}{2}.$$ In your case $ n= n_2 - n_1$.
Solution 2:
Let $k=n_2-n_1$. the number of states between (and including) energy levels $n_1$ and $n_2$ is $k+1$. thus the number of spectral lines is the number of ways of choosing 2 states from the $k+1$ states, which is $${k+1\choose 2}=\frac{(n_2-n_1+1)(n_2-n_1)}{2}$$