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If $n _{1}, n_{2}$ and $n _{3}$ are the fundamental frequencies of three segments into which a string is divided, then the original fundamental frequency $n$ of the string is given by
$n=n_{1}+n_{2}+n_{3}$
$\sqrt{n}=\sqrt{n_{1}}+\sqrt{n_{2}}+\sqrt{n_{3}}$
$\frac{1}{n}=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\frac{1}{n}$
$\frac{1}{\sqrt{n}}=\frac{1}{\sqrt{n}}+\frac{1}{\sqrt{n_{2}}}+\frac{1}{\sqrt{n_{3}}}$
Solution
Let $l$ be the length of the string.
Fundamental frequency is given by
$n=\frac{1}{2 l} \sqrt{\frac{T}{\mu}}$
$n \propto \frac{1}{l}(\because T \text { and } \mu \text { are constants })$
Here, $l_{1}=\frac{k}{n_{1}}, l_{2}=\frac{k}{n_{2}}, l_{3}=\frac{k}{n_{3}}$ and $l=\frac{k}{n}$
But $l=l_{1}+l_{2}+l_{3}$
$\therefore \quad \frac{1}{n}=\frac{1}{n_{1}}+\frac{1}{n_{2}}+\frac{1}{n_{3}}$
