Question

Sol. [Key Concept: Wave number $\propto \ Z^{2}(1/n_{1}^{2} - 1/n_{2}^{2})$; take Balmer's first line ($2 \rightarrow 3$) over Brackett's first ($4 \rightarrow 5$).]

$\ {\overline{V}}_{1} = R_{H}(z)^{2}\left\lbrack \frac{1}{2^{2}} - \frac{1}{3^{2}} \right\rbrack \Rightarrow I^{st}$ line of Balmer series ${\overline{V}}_{2} = R_{H}(z)^{2}\left\lbrack \frac{1}{4^{2}} - \frac{1}{5^{2}} \right\rbrack \Rightarrow I^{t}$ line of Brackett series $\frac{{\overline{V}}_{1}}{{\overline{\text{ }V}}_{2}} = \frac{500}{81} \Rightarrow 5:0.81$