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

$\frac{\text{sum of "}n\text{"observation}}{\text{Number of observatin }}=\overline{x}$

Sum of $"n"$ observation $=n\overline{x}$

After Add 1 to the 1^st her , 2 to the 2^nd here …$n$  to the $n^{th}$  term

Sum of observations $=$  Sum of  $"n"$ observation $+1+2+3+\cdots +n$

$=$  sum of $"n"$ observation $+\frac{n\left(n+1\right)}{2}$

$\therefore$ Mean $=$  $\frac{\text{sum of "}n\text{" observation}+n\left(n+1\right)}{2}$

$=\frac{\text{sum of "}n\text{"observatin }}{n}+\frac{\frac{n\left(n+1\right)}{2}}{n}$

$=\overline{x}+\left(\frac{n+1}{2}\right)$