When an electron is excited to $n^{th}$ energy state in hydrogen, the possible number of spectral lines emitted are 

If electron jumps from $n_2$ level to $n_1$ level by emitting energy, Then the spectrum formed is emission spectrum.
No. of emission spectrum lines $=\left[\frac{\left({\mathrm{n}}_2-{\mathrm{n}}_1\right)\left({\mathrm{n}}_2-{\mathrm{n}}_1+1\right)}{2}\right]$
When the value of $n_1=1$; and $n_2=n$;
$$\begin{gathered} \text{ No. of emission spectrum lines }=\frac{(n-1)(n-1+1)}{2} \\ =\frac{n(n-1)}{2}=\frac{n^2-n}{2} \end{gathered}$$