The mean of n observation is x.


The mean of $n$ observation is $\bar{x}$. If the first observation is increased by 1 , the second by 2 , the third by 3 , and so on, then the new mean is

(a) $\bar{x}+(2 n+1)$

(b) $\bar{x}+\frac{n+1}{2}$

(c) $\bar{x}+(n+1)$

(d) $\bar{x}-\frac{n+1}{2}$


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, the second by 2, the third by 3 and so on.

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

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


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


Hence, the correct answer is (c).


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