The weighted means of first n natural numbers whose weights are equal to the squares of corresponding numbers is
n+12
3n(n+1)2(2n+1)
(n+1)(2n+1)6
n(n+1)2
The required mean is given by
X¯=1⋅12+2⋅22+3⋅32+…+n⋅n212+22+…+n2=Σn3Σn2
⇒ X¯=n (n+1) 22n(n+1)(2n+1)6=3n(n+1)2(2n+1)