If ${\mathrm{n}}_1, {\mathrm{n}}_2 \mathrm{and} {\mathrm{n}}_3$, 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

Total length of string $\mathrm{l} = {\mathrm{l}}_1+{\mathrm{l}}_2+{\mathrm{l}}_3$
(As string is divided into three segments)

${\mathrm{n}}_{1 } = \frac{1}{2{\mathrm{l}}_1}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}; {\mathrm{n}}_2 = \frac{1}{2{\mathrm{l}}_2}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}; {\mathrm{n}}_3 = \frac{1}{2{\mathrm{l}}_3}\sqrt{\frac{\mathrm{T}}{\mathrm{\mu }}}$

Hence frequency $\propto$ $\frac{1}{\mathrm{length}}(\because \mathrm{n} = \frac{1}{2\mathrm{l}}\sqrt{\frac{\mathrm{T}}{\mathrm{m}}})$

$\therefore \frac{1}{\mathrm{n}}= \frac{2\mathrm{l}}{\sqrt{{\displaystyle \frac{\mathrm{T}}{\mathrm{\mu }}}}} = \frac{2{\mathrm{l}}_1}{\sqrt{{\displaystyle \frac{\mathrm{T}}{\mathrm{\mu }}}}}+\frac{2{\mathrm{l}}_2}{\sqrt{{\displaystyle \frac{\mathrm{T}}{\mathrm{\mu }}}}}+\frac{2{\mathrm{l}}_3}{\sqrt{{\displaystyle \frac{\mathrm{T}}{\mathrm{\mu }}}}} = \frac{1}{{\mathrm{n}}_1}+\frac{1}{{\mathrm{n}}_2}+\frac{1}{{\mathrm{n}}_3}$