c
(c) \(\frac{1}{{{k_{eff}}}} = \frac{1}{k} + \frac{1}{{2\,k}} + \frac{1}{{4\,k}} + \frac{1}{{8\,k}} + ....\)

\( = \frac{1}{k}\left[ {1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + .....} \right]\)

\( = \frac{1}{k}\left( {\frac{1}{{1 - 1/2}}} \right)\)

\( = \frac{2}{k}\) (By using sum of infinite geometrical progression \(a + \frac{a}{r} + \frac{a}{{{r^2}}} + ...\infty \) sum \((S)\) \( = \frac{a}{{1 - r}}\))

\(\therefore {k_{eff}} = \frac{k}{2}.\)