Infinite springs with force constants k , 2 k , 4 k a n d 8 k … . . respectively are connected in series. The effective force constant of the spring will be

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Effective force constant is equal to the reciprocal of the sum of individual force constant, hence

$\frac{1}{k_{e}} = \frac{1}{k_{1}} + \frac{1}{k_{2}} + \frac{1}{k_{3}} + \dots$

Given, $k_{1} = k, k_{2} = 2 k, k_{3} = 3 k, \dots . .$

$∴$ $\frac{1}{k_{e}} = \frac{1}{k} + \frac{1}{2 k} + \frac{1}{4 k} + \frac{1}{8 k} + \dots$

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.

$\Rightarrow$ $\frac{1}{k_{e}} = \frac{1}{k} \times \frac{1}{(1 - \frac{1}{2})} = \frac{2}{k} \Rightarrow k_{e} = \frac{k}{2}$

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