When a string of length ' \(l\) ' is divided into three segments of length \(l_1, l_2\) and \(l_3\). The fundamental frequencies of three segments are \(\mathrm{n}_1, \mathrm{n}_2\) and \(\mathrm{n}_3\) respectively. The original fundamental frequency ' \(n\) ' of the string is

The fundamental frequency of a string is given by
Given: \(l=l_1+l_2+l_3\)... (i)
\(\mathrm{n}=\frac{1}{2 l} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}\)
\(\Rightarrow \mathrm{n} \propto \frac{1}{l}\) or \(\mathrm{n} l=\mathrm{k}\)
\(\therefore \quad l_1=\frac{\mathrm{k}}{\mathrm{n}_1}, l_2=\frac{\mathrm{k}}{\mathrm{n}_2}\) and \(l_3=\frac{\mathrm{k}}{\mathrm{n} 3}\)... (ii)
\(\therefore \quad\) Original length \(l=\frac{\mathrm{k}}{\mathrm{n}}\)... (iii)
Putting eq (ii) and (iii) into eq (i)
\(\frac{\mathrm{k}}{\mathrm{n}}=\frac{\mathrm{k}}{\mathrm{n}_1}+\frac{\mathrm{k}}{\mathrm{n}_2}+\frac{\mathrm{k}}{\mathrm{n}_3}\)
\(\therefore \quad \frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}\)