Question

If the mean deviation of number $1,1+d,1+2d....,1+100d$from their mean is $255$, then $d$ is equal to

• A. $10.0$
• B. $20.0$
• C. $10.1$
• D. $20.2$

Answer 1

The correct option is C

$10.1$

Explanation of the correct option.

Compute the required value.

There are 51 terms in the series,

Thus mean $\left(\overline{x}\right)={51}^{st}term$

$\Rightarrow$ $\left(\overline{x}\right)=1+50d$

Since Mean deviation $\left(M.D.\right)$ is given by,

$$\begin{gathered} M.D.=\frac{1}{n}\sum _{i=1}^{101}\left|x_i-\overline{x}\right| \\ \Rightarrow 255=\frac{1}{101}\left[50d+49d+48d+..........+d+0++d+2d+......+50d\right] \\ \Rightarrow 255=\frac{1}{101}.2d(1+2+.....+50) \\ \Rightarrow 255=\frac{1}{101}.2d.\frac{50.51}{2} \\ \Rightarrow 255=\frac{50.51}{101}d \\ \Rightarrow d=\frac{101\times 255}{50\times 51} \\ \Rightarrow d=\frac{101\times 255}{2550} \\ \Rightarrow d=10.1 \end{gathered}$$

Hence option $\left(C\right)$ is the correct option.