$\begin{array}{lllllllll}{x}_i& :& 1^2& 2^2& 3^2& 4^2& ...& {\left(n-1\right)}^2& n^2\\ f_i& :& n& n-1& n-2& n-3& ...& 2& 1\end{array}$
$\therefore \sum _{i=1}^n{x}_i{f}_i=1^2\cdot n+2^2\left(n-1\right)+3^2\left(n-2\right)+...$$+{\left(n-1\right)}^{2}x+n^2\cdot 1$
$=\sum _{r=1}^n\left(n+1\right)r^2-r^3$
$=\left(n+1\right)\sum _{r=1}^n{r}^2-\sum _{r=1}^n{r}^3$

$=\frac{n{\left(n+1\right)}^2\left(2n+1\right)}{6}-\frac{n^2{\left(n+1\right)}^2}{4}$
$\frac{n{\left(n+1\right)}^2}{12}\left\{2\left(2n+1\right)-3n\right\}=\frac{n{\left(n+1\right)}^2\left(n+2\right)}{12}$
Also, $\mathrm{\Sigma }{f}_i=1+2+....+n=\frac{n\left(n+1\right)}{2}$
Mean $=\frac{\mathrm{\Sigma }{f}_i{x}_i}{\mathrm{\Sigma }{f}_i}=\frac{n{\left(n+1\right)}^2\left(n+2\right)}{12}\times \frac{2}{n\left(n+1\right)}$

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