The mean of n observation is X
Question:

The mean of $n$ observation is $\bar{X}$. If the first item is increased by 1 , second by 2 and so on, then the new mean is

(a) $\overline{\mathrm{X}}+n$

(b) $\overline{\mathrm{X}}+\frac{n}{2}$

(c) $\overline{\mathrm{X}}+\frac{n+1}{2}$

(d) None of these

Solution:

Let $x_{1}, x_{2}, x_{3}, \ldots, x_{n}$ be the $n$ observations.

Mean $=\bar{X}=\frac{x_{1}+x_{2}+\ldots+x_{n}}{n}$

$\Rightarrow x_{1}+x_{2}+x_{3}+\ldots+x_{n}=n \bar{X}$

If the first item is increased by 1, second by 2 and so on.

Then, the new observations are $x_{1}+1, x_{2}+2, x_{3}+3, \ldots, x_{n}+n$.

New mean $=\frac{\left(x_{1}+1\right)+\left(x_{2}+2\right)+\left(x_{3}+3\right)+\ldots+\left(x_{n}+n\right)}{n}$

$=\frac{x_{1}+x_{2}+x_{3}+\ldots+x_{n}+(1+2+3+\ldots+n)}{n}$

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

$=\bar{X}+\frac{n+1}{2}$

Hence, the correct answer is (c).