\({n_1} = 1,\,{n_2} = \infty \)

\(\frac{1}{\lambda } = R{z^2}\left( {\frac{1}{{n_1^2}} - \frac{1}{{n_2^2}}} \right)\)

\( \Rightarrow \frac{1}{A} = {1^2}\,R\left( {\frac{1}{1} - \frac{1}{\infty }} \right) \Rightarrow \frac{1}{A} = R\)

Longest wavelength \(= 1^{st}\) line

\({n_1} = 3,\,{n_2} = 4\)

\(\frac{1}{\lambda } = R{z^2}\left( {\frac{1}{{{3^2}}} - \frac{1}{{{4^2}}}} \right) \Rightarrow \frac{1}{\lambda } = \frac{{R7}}{{36}}\)

\(R = \frac{1}{A}\)

\(\frac{1}{\lambda } = \frac{{\frac{1}{A} \times 7}}{{36}} \Rightarrow \frac{1}{\lambda } = \frac{7}{{36\,A}} \Rightarrow \lambda  = \frac{{36\,A}}{7}\)