The problem involves two stages of expansion:

Stage 1: Isothermal Expansion Process: $PV = \text{constant}$ (since $T$ is constant). Initial State: $P_1 = p, V_1 = V$. Final State: $V_2 = 2V$. Using Boyle's Law: $P_1V_1 = P_2V_2$ $p \cdot V = P_2 \cdot 2V \implies P_2 = \frac{p}{2}$.

Stage 2: Adiabatic Expansion Process: $PV^\gamma = \text{constant}$ (where $\gamma = 5/3$ for monoatomic gas). Initial State: $P_2 = \frac{p}{2}, V_2 = 2V$. Final State: $V_3 = 16V$. Using the adiabatic relation: $P_2V_2^\gamma = P_3V_3^\gamma$ $\frac{p}{2} (2V)^{5/3} = P_3 (16V)^{5/3}$ $P_3 = \frac{p}{2} \left( \frac{2V}{16V} \right)^{5/3} = \frac{p}{2} \left( \frac{1}{8} \right)^{5/3}$ Calculate $(1/8)^{5/3}$: Since $8 = 2^3$, $(1/2^3)^{5/3} = 1/2^5 = 1/32$. $P_3 = \frac{p}{2} \cdot \frac{1}{32} = \frac{p}{64}$.