So, we get lengths of two open pipes as $L _{1}=\frac{ v }{2 n _{1}}$ and $L _{2}=\frac{ v }{2 n _{2}}$ Now the pipes are join in series.

Fundamental frequency of new open pipe of length $L _{1}+ L _{2},$

$n =\frac{ v }{2\left( L _{1}+ L _{2}\right)}$

$n =\frac{ v }{2\left(\frac{ v }{2 n _{1}}+\frac{ v }{2 n _{2}}\right)}$

$n=\frac{1}{\frac{1}{n_{1}}+\frac{1}{n_{2}}}$

$\Rightarrow n =\frac{ n _{1} n _{2}}{ n _{1}+ n _{2}}$