Particles of masses $$m,2m,3m, ........ nm\,grams$$       are placed on the same line at distances $$l, 2l, 3l, ...... nl\,cm$$     from a fixed point. The distance of centre of mass of the particles from the fixed point in centimetres is

Solution :
$$\eqalign{ & {x_{cm}} = \frac{{m \times \ell + 2m \times 2\ell + 3m \times 3\ell + ... + nm \times n\ell }}{{m + 2m + 3m + .. + nm}} \cr & = \frac{{\left( {1 + 4 + 9 + .. + {n^2}} \right)\ell }}{{1 + 2 + .. + n}} \cr & = \frac{{\frac{{\ell n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6}}}{{\frac{{n\left( {n + 1} \right)}}{2}}} \cr & = \frac{{\left( {2n + 1} \right)\ell }}{3} \cr} $$