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(D) The gravitational intensity $I$ at point $O$ due to multiple particles is the sum of the individual intensities: $I = I_1 + I_2 + I_3 + I_4 + \dot

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$4G$

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(D) The gravitational intensity $I$ at point $O$ due to multiple particles is the sum of the individual intensities: $I = I_1 + I_2 + I_3 + I_4 + \dots$ $I = \frac{GM}{r_1^2} + \frac{GM}{r_2^2} + \frac{GM}{r_3^2} + \frac{GM}{r_4^2} + \dots$ Given $M = 3 \ kg$ and distances $r = 1, 2, 4, 8, \dots$ $I = GM \left[ \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{4^2} + \frac{1}{8^2} + \dots ight]$ $I = GM \left[ 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots ight]$ This is an infinite geometric series with first term $a = 1$ and common ratio $r = \frac{1}{4}$. The sum is $S = \frac{a}{1 - r} = \frac{1}{1 - 1/4} = \frac{1}{3/4} = \frac{4}{3}$. Substituting this back: $I = GM 	imes \frac{4}{3}$ $I = G 	imes 3 	imes \frac{4}{3} = 4G$.

Infinite particles,each of mass $3 \ kg$,are placed at distances of $1 \ m, 2 \ m, 4 \ m, 8 \ m, \dots$ from point $O$. What is the gravitational intensity at point $O$?

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