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

Solution:

Let ${\mathrm{x}}_1,{\mathrm{x}}_2,\dots {\mathrm{x}}_{\mathrm{n}}$ be the n items. 

Given, $\overline{\mathrm{x}}=\frac{{\mathrm{x}}_1+{\mathrm{x}}_2+\dots +{\mathrm{x}}_{\mathrm{n}}}{\mathrm{n}}$

$\begin{array}{c}\therefore \text{ New mean }=\frac{\left({\mathrm{x}}_1+1\right)+\left({\mathrm{x}}_2+2\right)+\dots +\left({\mathrm{x}}_{\mathrm{n}}+\mathrm{n}\right)}{\mathrm{n}}\\ =\frac{\left({\mathrm{x}}_1+{\mathrm{x}}_2+\dots +{\mathrm{x}}_{\mathrm{n}}\right)+(1+2+\dots +\mathrm{n})}{\mathrm{n}}\\ =\overline{\mathrm{x}}+\frac{\mathrm{n}(\mathrm{n}+1)}{2\mathrm{n}}[\because \text{ from Eq. (i) }]\\ =\overline{\mathrm{x}}+\frac{(\mathrm{n}+1)}{2}\end{array}$