Let the terms are $x_1, x_2, x_3, \ldots x_n$
$\therefore \quad x_1+x_2+x_3+\ldots+x_n=n \bar{X}$
Now first term is increased by 1 , second by 2 and so on
$\therefore \quad x_1+1+x_2+2+\ldots+x_n+n=n \bar{X}+\frac{n(n+1)}{2}$
$\therefore \quad$ New mean $=\frac{n \bar{X}+\frac{n(n+1)}{2}}{n}=\bar{X}+\frac{(n+1)}{2}$
So, The correct option of this question will be (C).