How does pressure increase in a multistage compressor?
Firstly, your intuition doesn't really apply in terms of basic thermodynamics (but, as I'll explain, you're kind of right):


  Boyle's Law: "For a fixed mass of gas, at a constant temperature, the product (pressure x volume) is a constant."


Pressure does increase linearly [assuming the temperature stays the same]: if you squash an amount of a gas into half the volume, the pressure will double (assuming you keep the temperature the same). If you squash it into half the volume again, the pressure will double again (4x the original)

However you're also right in that we don't simply increase pressure by the same amount as we go along: Each stage compresses the air by a different amount (otherwise an engine would look like a weird triangle). and perhaps more importantly we can't maintain a constant temperature in an engine... as we compress the air into a smaller volume the temperature increases unless the air is being cooled in some way.

Essentially, though, you can calculate the pressure change at each stage if you know how much the volume reduces by (ie how much smaller each chamber is than the last) and the temperature of the air in each.