Explanation

(b)
The mean of the series $a,a + d,\ldots..,a + 2nd\$is
$\overline{x} = \frac{1}{2n + 1}\lbrack a + a + d + a + 2d + \ldots + a + 2nd\rbrack$
$= \frac{1}{2n + 1}\left\lbrack \frac{2n + 1}{2}(a + a + 2nd) \right\rbrack = a + nd$
$\therefore Mean$ deviation from mean
$= \frac{1}{2n + 1}\sum_{r = 0}^{2n}\left| (a + rd) - (a + nd) \right|$
$= \frac{1}{2n + 1}\sum_{r = 0}^{2n}{(r - n)d}$
$= \frac{1}{2n + 1}2d(1 + 2 + \ldots + n)$
$= \frac{n(n + 1)}{2n + 1}d$