Question

How many emission spectral lines are possible when hydrogen atom is excited to $n^{th}$ energy level?

• A. $\frac{n(n+1)}{2}$
• B. $\frac{(n+1)}{2}$
• C. $\frac{(n-1)n}{2}$
• D. $\frac{n^2}{4}$

Answer 1

The correct option is C $\frac{(n-1)n}{2}$

Thus the number of lines emitted from nth energy level:

$=1+2+3+......(n-1)$

$=\sum (n-1)$

As we know that, $\sum n=\frac{n(n+1)}{2}$

$\therefore \sum (n-1)=\frac{(n-1)(n-1+1)}{2}$

$=\frac{(n-1)n}{2}$

The number of spectral lines that appear in the hydrogen spectrum when an electron jumps from nth energy level$=\frac{n(n-1)}{2}.$

Hence, option C is correct.