Masses m 1 3 N 1   N are placed at x = N , when N = 2 , 3 , 4 , … ∞ . If the total mass of…

The arrangement of masses will look as given below,

Centre of the mass of the system along X-axis will be given as,

$X_{cm}=\frac{m_1{x}_1+m_2{x}_2+...}{m_1+m_2+...}$

$=\frac{\left[m{\left({\displaystyle \frac{1}{3}}\right)}^2{\displaystyle \frac{1}{2}}\right]\times 2+\left[m{\left(\frac{1}{3}\right)}^3\frac{1}{3}\right]\times 3+...}{M}$

$=\frac{m}{M}\times \left[{\left(\frac{1}{3}\right)}^2+{\left(\frac{1}{3}\right)}^3+...\right]$

$=\frac{m}{M}\times \frac{{\left({\displaystyle \frac{1}{3}}\right)}^2}{1-\left({\displaystyle \frac{1}{3}}\right)}$

$=\frac{m}{6M}$