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 $=\frac{\left(x_{1}+1\right)+\left(x_{2}+2\right)+\left(x_{3}+3\right)+\ldots+\left(x_{n}+n\right)}{n}$
$=\frac{x_{1}+x_{2}+x_{3}+\ldots+x_{n}+(1+2+3+\ldots+n)}{n}$
$=\frac{n \bar{x}+\frac{n(n+1)}{2}}{n}$
$=\bar{x}+\frac{n+1}{2}$
Hence, the correct answer is (c).