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

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

A
$\frac{n + 1}{2}$
B
$\frac{3 n (n + 1)}{2 (2 n + 1)}$
C
$\frac{(n + 1) (2 n + 1)}{6}$
D
$\frac{n (n + 1)}{2}$

Solution:

The required mean is given by

$\bar{X} = \frac{1 \cdot 1^{2} + 2 \cdot 2^{2} + 3 \cdot 3^{2} + \dots + n \cdot n^{2}}{1^{2} + 2^{2} + \dots + n^{2}} = \frac{Σ n^{3}}{Σ n^{2}}$

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

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