Solution

Everty time the pump is connected to the container and the piston is pulled, the volume of the gas increases from $V_0$ to $V_0+V$, and its pressure becomes $V_0/(V_0+V)$ of its former value (assuming constant temperature). In order for the pressure of the gass to drop from $P_0$ to $P$, we must apply $n$ strokes of the pump so that

$$\left(\frac{V_0}{V_0+V}\right)^n=\frac{P}{P_0}\quad\Rightarrow\quad n=\frac{\ln(P_0/P)}{\ln(1+V/V_0)}.$$