Explanation

(c)
The given series is 1,2,3,…(2n+1)
$\overline{x} = \frac{1 + 2 + 3 + \ldots + (2n + 1)}{2n + 1}$
=$\frac{(2n + 1)(2n + 2)}{2(2n + 1)}$
$= (n + 1)$
$\therefore\sigma^{2} = \frac{1}{2n + 1}\sum_{r = 0}^{2n}\left\{ (1 + r) - (1 + n) \right\}^{2}$
$= \frac{2}{2n + 1}\left( 1^{2} + 2^{2} + \ldots + n^{2} \right)$
${\Rightarrow \sigma}^{2} = \frac{n(n + 1)}{3}$
$\Rightarrow \ \ \ \sigma = \sqrt{\frac{n(n + 1)}{3}}$