If n₁, n₂ and n₃ are the fundamental frequencies of three elements into which a string is divided, then the original fundamental frequency n of the string is given by

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

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

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

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

answer is A.

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Detailed Solution

Total length of string, $l = l_{1} + l_{2} + l_{3}$
(As string is divided into three segments)
But frequency $\propto \frac{1}{length} (∵ f = \frac{1}{2 l} \sqrt{\frac{T}{m}})$
So, $\frac{1}{n} = \frac{1}{n_{1}} + \frac{1}{n_{2}} + \frac{1}{n_{3}}$