Question

If the mean of the set of numbers $x_1,x_2,....x_n$ is $\overline{x}$, then the mean of the numbers $x_i+2i,1\le i\le n$ is

• A. $\overline{x}+2n$
• B. $\overline{x}+n+1$
• C. $\overline{x}+2$
• D. $\overline{x}+n$

Answer 1

The correct option is B

$\overline{x}+n+1$

$\overline{x}=\frac{{\sum }_{i=1}^n{x}_i}{n}\Rightarrow {\sum }_{i=1}^n{x}_i=n\overline{x}$
$\therefore \frac{{\sum }_{i=1}^n(x_i+2i)}{n}=\frac{{\sum }_{i=1}^n{x}_i+2(1+2+...+n)}{n}=\frac{n\overline{x}+2\frac{n(n+1)}{2}}{n}=\overline{x}+(n+1)$