${\mathrm{\nu }}_1$ and ${\mathrm{\nu }}_2$ are the frequencies of  limiting line  for Lyman and Balmer series. Which relation is correct for frequency ${\mathrm{\nu }}_3$ of 1^st line of Lyman series?

 Given, frequencies of  limiting line of Lyman and Balmar series are as v₁ and v₂ respectively

Thus${\mathrm{v}}_1=\mathrm{hc}{\overline{\mathrm{v}}}_1={\mathrm{hcR}}_{\mathrm{H}}\left(\frac{1}{1^2}-\frac{1}{{\infty }^2}\right)$ = hcR_H

${\mathrm{v}}_2=\mathrm{hc}{\overline{\mathrm{v}}}_2={\mathrm{hcR}}_{\mathrm{H}}\left(\frac{1}{2^2}-\frac{1}{{\infty }^2}\right)$$=\frac{\mathrm{hcRH}}{4}$

for the first line in Lyman series

${\mathrm{v}}_3=\mathrm{hc}{\overline{\mathrm{v}}}_3={\mathrm{hcR}}_{\mathrm{H}}\left(\frac{1}{1^2}-\frac{1}{2^2}\right)$$=\frac{3{\mathrm{hcR}}_{\mathrm{H}}}{4}$

Thus ${\mathrm{v}}_3={\mathrm{v}}_1-{\mathrm{v}}_2$