Your "1/4" and "3/16" are correct but you have the wrong pattern after that. After removing 1/4+ 3/16= 7/16 of the you have 9/16 left and so the third stroke remove 1/4(9/16)= 9/64, not 3/64.

I think it is simpler to calculate how much is left after each pump. After the first pump, 1/4 of the air is removed leaving 3/4, 1/4 of that is 3/16 leaving 9/16, 1/4 of that is 9/64 leaving 27/64, etc. You should be able to see that you are getting powers of 3/4. After n pumps, of the air is left.