Let, \(F_1, F_2, F_4, F_8 \ldots \ldots\) be the forces of gravitation due masses ' \(m\) ' at \(x=1,2,4,8 \ldots\) respectively.

\(\Rightarrow F_1=\frac{G m^2}{1^2}\)

\(F_2=\frac{G m^2}{2^2}\)

\(F_4=\frac{G m^2}{4^2}\)

\(F_8 =\frac{G m^2}{8^2}\)

\(F_1+F_2+F_4+F_8 \ldots=G m^2\left(\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+\frac{1}{64} \ldots\right)\)

infinite \(G.P\). with common ratio \(=\frac{1}{4}\)

For an infinite \(G.P\), sum \(=\left(\frac{a}{1-r}\right)\)

\(a\) is the first term

\(r\) is the common ratio

\(\Rightarrow \text { Sum }=\frac{1}{1-\frac{1}{4}}=\left(\frac{4}{3}\right)\)

\(\Rightarrow F_1+F_2+F_4+F_8 \ldots \ldots=\frac{4}{3} G m^2\)