The mean deviation from mean of the observations a,a+d,a+2d,…,a+2nd is

A
$\frac{n (n + 1) d^{2}}{3}$
B
$\frac{n (n + 1)}{2} d^{2}$
C
$a + \frac{n (n + 1) d^{2}}{2}$
D
None of these

Solution:

$\begin{array}{l} \bar{x} = \frac{1}{2 n + 1} [a + (a + d) + \dots + (a + 2 nd)] \\ = \frac{1}{2 n + 1} [(2 n + 1) a + d (1 + 2 + \dots + 2 n)] \\ = (a + d \frac{2 n}{2}) \frac{(2 n + 1)}{2 n + 1} = a + nd \\ ∴ MD from mean = \frac{1}{2 n + 1} Σ |x_{i} - \bar{x}| \\ = \frac{1}{2 n + 1} 2 | d | (1 + 2 + \dots + n) \\ = \frac{n (n + 1) | d |}{(2 n + 1)} \end{array}$

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