The weighted means of first n natural numbers whose weights are equal to the squares of corresponding numbers is

Solution:

The required mean is given by

$\overline{X}=\frac{1\cdot 1^2+2\cdot 2^2+3\cdot 3^2+\dots +n\cdot n^2}{1^2+2^2+\dots +n^2}=\frac{\mathrm{\Sigma }{n}^3}{\mathrm{\Sigma }{n}^2}$

$\Rightarrow \overline{X}=\frac{{\left\{\frac{n^{ }(n+1{)}^{ }}{2}\right\}}^2}{\frac{n(n+1)(2n+1)}{6}}=\frac{3n(n+1)}{2(2n+1)}$