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# If the mean of the set of numbers $x_1, x_2.....x_n$ is $\overline x$ then mean of the numbers $x_i+2i \: \: 1 \leq i \leq n$ is

$\begin {array} {1 1} (A)\;\overline x + 2n & \quad (B)\;\overline x +n+1 \\ (C)\;\overline x +2 & \quad (D)\;\overline x+n \end {array}$

Mean = $\large\frac{1}{n}$ $\sum_{i=1}^{n} (x_i+2_i)$
$= \large\frac{1}{n}$ $(x_1+2)+(x_2+4)+(x_3+6)+.....+(x_n+2n)$
= $\large\frac{1}{n}$ $(x_1+x_2+....+x_n)+\large\frac{1}{n}$ $(2+4+6+.....+2n)$
$= \overline x + \large\frac{2}{n}$. $\large\frac{n(n+1)}{2}$
$= \overline x + n+1$
Ans : (B)