JEE Main · 2023 · 15 Apr 2023 · Shift 1 · Modern Physics

The levels A, B and C are the first, second and third excited states, that is $n=2$, $n=3$ and $n=4$. For the transition giving ${\lambda }_1$ (level B to level A), $\frac{1}{{\lambda }_1}=R\left(\frac{1}{2^2}-\frac{1}{3^2}\right)=\frac{5R}{36}$. For ${\lambda }_2$ (level C to level B), $\frac{1}{{\lambda }_2}=R\left(\frac{1}{3^2}-\frac{1}{4^2}\right)=\frac{7R}{144}$. Dividing, $\frac{{\lambda }_1}{{\lambda }_2}=\frac{7/144}{5/36}=\frac{7}{20}=\frac{7}{4\times 5}$. Comparing with $\frac{7}{4n}$, the value of $n$ is $5$.