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(B) The gravitational potential $V$ at the origin due to a system of point masses is given by the sum of potentials due to individual masses:$V = - \l

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$ - 2Gm$

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(B) The gravitational potential $V$ at the origin due to a system of point masses is given by the sum of potentials due to individual masses:$V = - \left[ \frac{Gm}{r_1} + \frac{Gm}{r_2} + \frac{Gm}{r_3} + \dots \right]$Substituting the given positions $r_1 = 1, r_2 = 2, r_3 = 4, r_4 = 8, \dots$:$V = - Gm \left[ \frac{1}{1} + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots \right]$The term in the bracket is an infinite geometric series with first term $a = 1$ and common ratio $r = 1/2$. The sum of an infinite geometric series is $S = \frac{a}{1 - r}$.$S = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2$Therefore,$V = - Gm(2) = - 2Gm$.

Objects of mass $m$ are placed on the $x$-axis at $x = 1, x = 2, x = 4, x = 8, \dots$ respectively. What is the gravitational potential at the origin $x = 0$?

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