When `n` number of particles of masses `m, 2m, 3m, …….. nm` are at distances `x_(1)=1, x_(2)=4, x_(3)=9, ………x_(n)=n^(2)` units respectively from origin on the x-axis, then find the distance of their centre of mass from origin.

Explanation

`x_(cm)=(m(1)+2m(4)+3m(9)+.........+nm(n^(2)))/(m+2m+3m+......+nm)`
`=(m(1+2^(3)+3^(3)+......+n^(3)))/(m(1+2+3+.......+n))`
`=(((n(n+1))/(2))^(2))/((n(n+1))/(2))=(n(n+1))/(2)`