Question

If the mean of the set of numbers $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ is $\bar{x}$, then the mean of the numbers $x_{i}+2 i, 1 \leq i \leq n$ is

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

Answer 1

Answer: (B)

We know that $\bar{x}=\frac{\displaystyle\sum_{i=1}^{n} x_{i}}{n}$
$ \Rightarrow \displaystyle\sum_{i=1}^{n} x_{i}=n \bar{x}$
$\therefore \frac{\displaystyle\sum_{i=1}^{n}\left(x_{i}+2 i\right)}{n}=\frac{\displaystyle\sum_{i=1}^{n} x_{i}+2 \displaystyle\sum_{i=1}^{n} i}{n}=\frac{n \bar{x}+2(1+2+\ldots n)}{n}$
$=\frac{n \bar{x}+2 \frac{n(n+1)}{2}}{n}$
$=\bar{x}+(n+1)$