If λ_1 and λ_2 are the wavelengths of the first members of the...

For first line of Lyman series, $n_1=1 and$n_2=2 \therefore \quad \frac{1}{\lambda_1}=R\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=R\left(1-\frac{1}{4}\right)=\frac{3 R}{4}$For first line of Paschen series$n_1=3$and$n_2=4 \therefore \quad \frac{1}{\lambda_2}=R\left(\frac{1}{3^2}-\frac{1}{4^2}\right)=R\left(\frac{1}{9}-\frac{1}{16}\right)$=\frac{7 R}{144}$ $\therefore \quad \frac{\lambda_1}{\lambda_2}=\frac{7 R}{144} \times \frac{4}{3 R}=\frac{7}{108}$