If n₁, n₂  and n₃  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 following :

Concept:

Fundamental frequency:

The lowest frequency of any vibrating object is called a fundamental frequency.

The fundamental frequency of a string is given by

\(n = \frac{1}{{2l}}\;\sqrt {\frac{T}{m}}\;\)

Where l = length of the string, T = tension on the string and m =  linear mass density

​Calculation:

Given:

The fundamental frequency of a string (l) is given by:

\(⇒ n = \frac{1}{{2l}}\;\sqrt {\frac{T}{m}} \)

As T and m is constant

\(\therefore n\propto \frac {1}{l}\)

⇒ n1l1 = n2l2 = n3l3 = k        [Where k = constant]

\(⇒ l_1=\frac{k}{n_1},\,\,\, l_2=\frac{k}{n_2},\,\,\, l_3=\frac{k}{n_3}\)

The original length of the string is

\(\Rightarrow l=\frac{k}{n}\)

The total length of the string is 

⇒ l = l1 + l2 + l3

Substitute the value of l, l1, l2, and l3 in the above equation, we get

\(\Rightarrow \frac{k}{n} = \frac{k}{n_1}+\frac{k}{n_2}+\frac{k}{n_3}\)

\(\Rightarrow \frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}\)