Solution:

For Brackett series, \(n_1 = 4\). The wave number formula is \(bar{\nu} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)\). Thus, \(frac{9}{400}R = R \left( \frac{1}{16} - \frac{1}{n_2^2} \right) ⇒ \frac{1}{n_2^2} = \frac{1}{16} - \frac{9}{400} = \frac{16}{400} = \frac{1}{25} ⇒ n_2 = 5\).