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# Infinite numbers of masses, each of mass m,are placed along straight line at distances of r,2r,4r,8r etc from a reference point O.Find the gravitational field at point O

$(a)\;\frac{2GM}{r^2} \quad (b)\;\frac{4GM}{3r^2} \quad (c)\;\frac{4GM}{r^2} \quad (d)\;\frac{2GM}{3r^2}$

The magnitude of gravitational field intensity at point O will be the sum of gravitational field intensity due to each mass m
$I=Gm\bigg[\large\frac{1}{r^2}+\frac{1}{(2r)^2}+\large\frac{1}{(4r^2)}....\bigg]$
$\quad=\large\frac{Gm}{r^2}$$\bigg[1+\large\frac{1}{2^2}+\frac{1}{4^2}+\large\frac{1}{8^2}....\bigg]$
$\quad=\large\frac{Gm}{r^2}\bigg[\large\frac{1}{1-\Large\frac{1}{2^2}}\bigg]$
$\quad=\large\frac{Gm}{r^2}\bigg[\large\frac{1}{1-\Large\frac{1}{4}}\bigg]$
$\quad=\large\frac{4Gm}{3r^2}$
Hence b is the correct answer.
edited Jun 30, 2014
How that series is solved?