According to Bohr atom model, in which of the following transitions will the frequency be maximum?

According to the Bohr model, when an electron transitions from a higher energy level $$n_i$$ to a lower energy level $$n_f$$, the frequency of the emitted photon is given by $$\nu = \frac{E_i - E_f}{h} = \frac{13.6 \text{ eV}}{h}\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$.

For the transition $$n = 2 \to n = 1$$: $$\nu \propto \left(\frac{1}{1^2} - \frac{1}{2^2}\right) = 1 - \frac{1}{4} = \frac{3}{4} = 0.75$$.

For the transition $$n = 3 \to n = 2$$: $$\nu \propto \left(\frac{1}{4} - \frac{1}{9}\right) = \frac{5}{36} \approx 0.139$$.

For the transition $$n = 4 \to n = 3$$: $$\nu \propto \left(\frac{1}{9} - \frac{1}{16}\right) = \frac{7}{144} \approx 0.049$$.

For the transition $$n = 5 \to n = 4$$: $$\nu \propto \left(\frac{1}{16} - \frac{1}{25}\right) = \frac{9}{400} = 0.0225$$.

Comparing all values, the transition $$n = 2 \to n = 1$$ gives the maximum value of $$\frac{3}{4}$$, and hence the maximum frequency.

The correct answer is the transition from $$n = 2$$ to $$n = 1$$.