Why doesn't the apple pencil first generation stylus work with ipad air 3gen

Yes it does.

Are you sure you have the iPad Air 3rd generation?

Tell us the model number on your iPad.

Do you own the most current 10.5 inch screen iPad Air 3?

Tap and try all the troubleshooting tips in the Apple Pencil support link.

Use Apple Pencil with your iPad or iPad Pro - Apple Support

FWIW, I have an iPad Air 3 and the 1st generation Apple Pencil. The pencil works very well with my iPad.

Hello,

I suggest that you reach out to Apple for assistance.

Contact AppleCare

Try all of the troubleshooting tips in the above link I posted.