The mean of n observation is x.

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).