but when you extract air from APU to move the starter, and then it gear to the HP compressor... why dont give that air direct to the compressor to move it???
If you take the same amount of air as used in a small voulmn starter and deliver it to the compressor intake directly it would be wasted in the inefficiencies of blowing though stators at the wrong pressure conditions. If you've got lots of air to fill the whole compressor under pressure then you can windmill start it that way in flight.
So by using a small amount of air in a starter you can spin up the compressor high enough in speed and get only the rear smaller volumn stages to pressurize the burner.
maybe somebody else can untwist my words a bit here and explain it more clearly