a
\(\frac{1}{\lambda}= R \left[\frac{1}{ n _1{ }^2}-\frac{1}{ n _2{ }^2}\right]\) for \(H\)-atom

For balmer series, \(n _1=2\)

\(\frac{1}{\lambda}= R \left[\frac{1}{4}-\frac{1}{ n _2^2}\right]\)

For \(H _{ i }, n _2=3\)

\(H _\beta, n _2=4\)

\(\frac{1}{\lambda_{ H _{ \alpha }}}= R \left[\frac{1}{4}-\frac{1}{9}\right]=\frac{5 R }{36}\)

\(\frac{1}{\lambda_{ H _{ \beta }}}= R \left[\frac{1}{4}-\frac{1}{16}\right]=\frac{3 R }{16}\)

\(\frac{\frac{1}{\lambda_{H_\alpha}}}{\frac{1}{\lambda_{H_\beta}}}=\frac{\frac{5 R }{36}}{\frac{3 R }{16}}\)

\(\frac{\lambda_{H_{\alpha}}}{\lambda_{ H _{\beta}}}=\frac{27}{20}=\frac{ x }{20}\)

\(x=27\)