Question

The mean of n observation $\overrightarrow{x},$ if the first term is increased by 1 and second term is increased by 2 and so on The new mean is

• A. $x+\left(\frac{n-1}{2}\right)$
• B. $x+\left(\frac{n+1}{2}\right)$
• C. $x+\frac{n(n-1)}{2}$
• D. $x+n$

Answer 1

Answer: (B)

$x+\left(\frac{n+1}{2}\right)$

Mean of $x_1,x_2,...x_n$ is $x$

$\frac{x_1+x_2+x_3+...+x_n}{n}=x$

$\Rightarrow \sum _{i=1}^{n}x=nx$ __(1)

on adding $1$ to $x_1,2$ to $x_2$ and so on

we get $x_1+1+x_2+2+...+x_n+n$

$=(x_1+x_2+...+x_n)+(1+2+...+n)$

$=\sum n+\frac{n(n+1)}{2}$

New mean $=\frac{Newsummation}{n}$

$=\frac{\sum n+\frac{n(n+1)}{2}}{n}=\frac{2nx+n(n+1)}{2n}=x+\left(\frac{n+1}{2}\right)$