If $n_{1}, n_{2}, n_{3} ........$ are the frequencies of segments of a stretched string the frequency n of the string is given by ..

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$n = n_{1} + n_{2} + n_{3} + ........$

$n = \sqrt{n_{1} \times n_{2} \times n_{3} \times .....}$

$n = \sqrt{n_{1}^{2} + n_{2}^{2} + n_{3}^{2} + .....}$

$\frac{1}{n} = \frac{1}{n_{1}} + \frac{1}{n_{2}} + \frac{1}{n_{3}} + ...$

answer is C.

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Detailed Solution

For a vibrating string
$n_{1} l_{1} = n_{2} l_{2} = n_{3} l_{3} ...... = c o n s \tan t = k (s a y) = n l$
Also $l_{1} + l_{2} + l_{3} + l_{4} + ....... = 1$

$\frac{k}{n_{1}} + \frac{k}{n_{2}} + \frac{k}{n_{3}} + \frac{k}{n_{4}} + ...... = \frac{k}{n} \Rightarrow \frac{1}{n} = \frac{1}{n_{1}} + \frac{1}{n_{2}} + \frac{1}{n_{3}} + ...$