It is the gas generator that produces all the energy by continuous thermal reaction. Most of the thrust is produced by converting this energy to rotational force to drive the fan; some thrust is produced by the jet efflux from the gas generator. The difference between the two is the bypass ratio.
Well, that's close. Actually the bypass ratio is defined as the mass flow rate (i.e. kg/sec) through the fan nozzle divided by the mass flow rate through the core nozzle. Since the two are different temperatures and velocities, the thrust ratio is somewhat different. (For a typical high-bypass fan engine, the fan nozzle accounts for 75-80% of total thrust)