Infinite springs with force constant $k$, $2k$, $4k$ and $8k$.... respectively are connected in series. The effective force constant of the spring will be
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(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}}$)
$\therefore {k_{eff}} = \frac{k}{2}.$
$ = \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}}$)
$\therefore {k_{eff}} = \frac{k}{2}.$
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