If $n_1,n_2,n_3\mathrm{........}$  are the frequencies of segments of a stretched string the frequency n of the string is given by ..

For a vibrating string
$n_1{l}_1=n_2{l}_2=n_3{l}_3\mathrm{......}=cons\tan t=k\left(say\right)=nl$
Also  $l_1+l_2+l_3+l_4+\mathrm{.......}=1$

$\frac{k}{n_1}+\frac{k}{n_2}+\frac{k}{n_3}+\frac{k}{n_4}+\mathrm{......}=\frac{k}{n}\Rightarrow \frac{1}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}+\mathrm{...}$