b
\(\frac{1}{{{R_{eq}}}}\,\, = \,\,\frac{1}{R}\,\, + \;\,\frac{1}{{2R}}\,\, + \,\,\frac{1}{{4R}}\,\, + \;\,\frac{1}{{8R}}\,\,.........\infty \)

\( = \,\,\frac{1}{R}\,\,\left[ {1\,\, + \;\,\frac{1}{2}\,\, + \;\,\frac{1}{4}\,\, + \,\,......\infty } \right]\)

\( = \,\,\frac{1}{R}\,\,\left[ {\frac{1}{{1\,\, - \,\,\frac{1}{2}}}} \right]\,\, = \,\,\frac{2}{R}\)

\(\frac{1}{{{R_{eq}}}}\,\, = \,\,\frac{2}{R}\,\,\, \Rightarrow \,\,{R_{eq}}\,\, = \,\,\frac{R}{2}\)