If the variate of a distribution takes the values 1 ^ 2 , 2 ^ 2 , 3...

$\begin{cases}{x}_{i} & : & {1}^{2} & {2}^{2} & {3}^{2} & {4}^{2} & . . . & \left(n - 1\right)2 & n^{2} \\ {f}_{i} & : & n & n - 1 & n - 2 & n - 3 & . . . & 2 & 1\end{cases}$
$∴ \mathop{∑}_{i = 1}^{n}{x}_{i}{f}_{i} = {1}^{2} ⋅ n + {2}^{2}\left(n - 1\right) + {3}^{2}\left(n - 2\right) + . . .+ \left(n - 1\right)2x + n^{2} ⋅ 1$
$= \mathop{∑}_{r = 1}^{n}\left(n + 1\right)r^{2} - r^{3}$
$= \left(n + 1\right)\mathop{∑}_{r = 1}^{n}r^{2} - \mathop{∑}_{r = 1}^{n}r^{3}$

$= \dfrac{n\left(n + 1\right)2\left(2n + 1\right)}{6} - \dfrac{n^{2}\left(n + 1\right)2}{4}$
$\dfrac{n{\left(n + 1\right)}^{2}}{12}\left\{2\left(2n + 1\right) - 3n\right\} = \dfrac{n{\left(n + 1\right)}^{2}\left(n + 2\right)}{12}$
Also, $Σ{f}_{i} = 1 + 2 + . . . . + n = \dfrac{n\left(n + 1\right)}{2}$
Mean $= \dfrac{Σ{f}_{i}{x}_{i}}{Σ{f}_{i}} = \dfrac{n\left(n + 1\right)2\left(n + 2\right)}{12} \times \dfrac{2}{n\left(n + 1\right)}$

$= \dfrac{\left(n + 1\right)\left(n + 2\right)}{6}$