Another Proof by induction problem?

Well, if you look at it, A3 will locially be three, because the average of A1 and A2 is 3. So the average is obviously 3 now, and if you average a number (in this case 3) with a separate series of other numbers that already average your first number, the average will remain the same, because the number you are adding to the series does not pull the average at all away from its current state, so it will continuously remain with that average. Therefore, every following number will also be 3.