Question

The mean of n items is $\overline{x}.$ If the first term is increased by 1, the second by 2 and so on, then the new mean is .

• A. $\overline{x}+n$
• B. $\overline{x}+\frac{n}{2}$
• C. $\overline{x}+\frac{n+1}{2}$

Answer 1

The correct option is C $\overline{x}+\frac{n+1}{2}$

Let $x_1,x_2.......x_n$ be n items.
Then, $\overline{x}=\frac{1}{n}\sum x_i$
$Let{y}_1=x_1+1,y_2=x_2+2,y_3=x_3+3,.....,y_n=x_n+n$
Then the mean of the new series is:
$\frac{1}{n}\sum y_i=\frac{1}{n}\sum _{i=1}^n(x_i+i)$
$=\frac{1}{n}\sum _{i=1}^n{x}_i+\frac{1}{n}(1+2+3+......+n)$
$=\overline{x}+\frac{1}{n}.\frac{n(n+1)}{2}=\overline{x}+\frac{n+1}{2}$