Infinite number of bodies, each of mass $$2\,kg$$  are situated on $$X$$-axis at distances $$1\,m,2\,m, 4\,m$$   and $$8\,m,$$  respectively from the origin. The resulting gravitational potential due to this system at the origin will be

Solution :
The resulting gravitational potential,
$$\eqalign{ & V = - 2G\left[ {\frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ...} \right] \cr & \Rightarrow V = - 2G\left[ {1 + \frac{1}{2} + \frac{1}{{{2^2}}} + \frac{1}{{{2^3}}} + ...} \right] \cr & \Rightarrow V = - 2G{\left( {1 + \frac{1}{2}} \right)^{ - 1}} \cr & \Rightarrow V = - \frac{{2G}}{{\left( {1 - \frac{1}{2}} \right)}} = - \frac{{2G}}{{\left( {\frac{1}{2}} \right)}} = - 4G \cr} $$