To determine the value of $ n $, we start by considering the specific heat ratios for gases A and B.

For gas A, which is monoatomic with 3 translational degrees of freedom:

The degrees of freedom, $ f_A $, is 3.

Therefore, $ \gamma_A = \frac{f_A + 2}{f_A} = \frac{3 + 2}{3} = \frac{5}{3} $.

For gas B, which is polyatomic with 3 translational, 3 rotational degrees of freedom, and 1 vibrational mode:

The total degrees of freedom, $ f_B $, is $ 3 + 3 + 2 \times 1 = 8 $ (since one vibrational mode contributes 2 degrees of freedom: 1 kinetic + 1 potential).

Therefore, $ \gamma_B = \frac{f_B + 2}{f_B} = \frac{8 + 2}{8} = \frac{10}{8} = \frac{5}{4} $.

Now, the relation given is:

$ \frac{\gamma_A}{\gamma_B} = \left(1 + \frac{1}{n}\right) $

Substitute the values of $ \gamma_A $ and $ \gamma_B $:

$ \frac{\frac{5}{3}}{\frac{5}{4}} = \frac{5}{3} \times \frac{4}{5} = \frac{20}{15} = \frac{4}{3} $

Equating to the given form:

$ \frac{4}{3} = 1 + \frac{1}{n} $

Thus, we have:

$ \frac{4}{3} - 1 = \frac{1}{n} $

$ \frac{1}{3} = \frac{1}{n} $

Solving for $ n $:

$ n = 3 $

Hence, the value of $ n $ is 3.