According to Bohr's model of the hydrogen-like atom:

• Orbit Radius: $r_n = \frac{n^2 h^2 \varepsilon_0}{\pi m Z e^2} \propto n^2$ (Directly proportional to $n^2$, Matches I).
• Angular Momentum: $L = m v_n r_n = \frac{n h}{2\pi} \propto n$ (Directly proportional to $n$, Matches II).
• Orbital Velocity: $v_n = \frac{Z e^2}{2 \varepsilon_0 n h} \propto \frac{1}{n}$ (Inversely proportional to $n$, Matches III).
• Total Energy: $E_n = -\frac{m Z^2 e^4}{8 \varepsilon_0^2 n^2 h^2} \propto -\frac{1}{n^2}$ (Magnitude is inversely proportional to $n^2$, Matches IV).

Correct match: A-I, B-II, C-III, D-IV.

The shortest wavelength (series limit) for any spectral series corresponds to a transition from $n_2 = \infty$ to the lower energy level $n_1$:

• For Lyman series ($n_1 = 1$):

• For Paschen series ($n_1 = 3$):