Question

If the mean of the set of numbers $x_1,x_2,x_3,\dots ,x_n$ is $\bar{x}$, then the mean of the numbers $x_i+2i,1\le i\le n$ is

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

Answer 1

Answer: (B)

We know that $\bar{x}=\frac{\sum _{i=1}^n{x}_i}{n}$
$\Rightarrow \sum _{i=1}^n{x}_i=n\bar{x}$
$\therefore \frac{\sum _{i=1}^n\left(x_i+2i\right)}{n}=\frac{\sum _{i=1}^n{x}_i+2\sum _{i=1}^{n}i}{n}=\frac{n\bar{x}+2(1+2+\dots n)}{n}$
$=\frac{n\bar{x}+2\frac{n(n+1)}{2}}{n}$
$=\bar{x}+(n+1)$