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Q.
The frequencies for series limit of Balmer and Paschen series respectively are ' $v_{1}$' and ' $v_{3}$'. If frequency of first line of Balmer series is '$v_{2}$' then the relation between '$v_{1}', 'v_{2}$' and '$v _{3}$' is
Wavelength of Balmer series limit
$\frac{1}{\lambda_{1}}= R \left(\frac{1}{2^{2}}-\frac{1}{\infty}\right)$
Or $\frac{1}{\lambda_{1}}=\frac{ R }{4}$
Using $c=v \lambda$
We get frequency for Balmer series limit $\frac{v_{1}}{c}=\frac{ R }{4}$
$\Rightarrow v_{1}=\frac{ Rc }{4}\,\,\, ....$(1)
Wavelength of Paschen series limit $\frac{1}{\lambda_{3}}= R \left(\frac{1}{3^{2}}-\frac{1}{\infty}\right)$
Or $\frac{1}{\lambda_{3}}=\frac{ R }{9}$
We get frequency for Paschen series limit $v_{3}=\frac{ Rc }{9} \,\,\,\ldots$ (2)
Wavelength of first line of Balmer series $\frac{1}{\lambda_{2}}= R \left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right) $
where $c = v _{2} \lambda_{2}$
$\Rightarrow v_{2}=\text{Rc}\left(\frac{1}{4}-\frac{1}{9}\right) \,\,\, \ldots (3)$
From equations (1), (2) and (3) we get
$v_{2}=v_{1}-v_{3}$