Question

Let the mean of $n$ terms be $\overline{x}$, if the first term is increased by $1$, second term is increased by $2$ and so on, then the new mean is

• A. $\overline{x}+n$
• B. $\frac{\overline{x}+n}{2}$
• C. $\overline{x}+\left(\frac{n+1}{2}\right)$
• D. $\overline{x}+\left(\frac{n-1}{3}\right)$

Answer 1

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

$\overline{x}=\frac{\sum _{i=1}^n{x}_i}{n}$
New mean $=\frac{\sum _{i=1}^n\left(x_i+i\right)}{n}=\frac{\sum _{i=1}^n{x}_i}{n}+\frac{\sum _{i=1}^{n}i}{n}$

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