You are incorrect...and owe an apology to Atlantin. Here is why:
An object at rest has an average speed of 0 (zero). An object accelerating at one g (32 ft/sec) has a velocity of 32 feet/sec at the end of one second. IF (which is what you are assuming) the object immediately goes from 0 to 32 ft/sec instantly and maintains that speed for one second...then it will have moved 32 feet in that second. However, that is not what happens. It accelerates from 0 to 32 feet/sec in a matter of one second. That gives the object an average speed of 16 feet/sec for that period of one second. You cannot average 32 feet/sec when you start from zero and accelerate. So...after one second, you are traveling at 32 feet/sec but have averaged 16 feet/second over the course of that second...so you ahve traveled 16 feet in that second. Understand?
As far as the second second goes (this is easier to see)...the object is traveling at 64 feet/sec at the end of the 2 seconds. So...it started at 0, ended at 64 after 2. So...the average speed is (0+32+64)/2 = Average 32 ft/sec*2=64 feet after 2 seconds.
After 3 seconds: Starts at 0, ends at 96. Average speed over 3 seconds is 48 feet/sec for 3 seconds = 144 feet.
The key to understanding this is to remember that acceleration of 32 feet per second is a linear process...and is not "stepped" every second. In other words, the start is not zero and then at 1/10th of a second you are going 32 ft/sec, at 2/10th of a sec you are going 32 ft/sec...etc etc...to 10/10ths of a second you are going 32 ft/sec then at 11/10ths of a second (1 1/10 sec) you are going at 64 ft/sec. It is a straight acceleration from zero to 32 and you can draw a 45 degree line connecting the points. Linear.
Thanks, Nelson... I already have apologized to Atlantin. It was late and I had a senior moment... my brain was on vacation.