Explanation

(c) Effective force constant is equal to the reciprocal of the sum of individual force constant, hence
$\frac{1}{k_{e}} = \frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}} + \ldots$
Given, $k_{1} = k,\ k_{2} = 2k,\ k_{3} = 3k,\ldots..$
$\therefore$ $\frac{1}{k_{e}} = \frac{1}{k} + \frac{1}{2k} + \frac{1}{4k} + \frac{1}{8k} + \ldots$
The given series a geometric progression series, hence sum is
$S_{\infty} = \frac{a}{1 - r}$
where $a$ is first term of series and $r$ the common difference.
$\Rightarrow$ $\frac{1}{k_{e}} = \frac{1}{k} \times \frac{1}{\left( 1 - \frac{1}{2} \right)} = \frac{2}{k} \Rightarrow k_{e} = \frac{k}{2}$