simple acceleration question

Hi guys. I've got another simple physics question that I must understand wrong and I'm looking for some help to better explain it.

The acceleration of a moving object has a constant magnitude of a = 3.00 m/s^2. The object starts out from a dead stop at t = 0.00 s and moves in a straight-line path. How far will it have traveled from its starting point at t = 5.00 s?

I think to myself that this object will have a speed of 0 at t=0, 3 at t=1, 6 at t=2, 9 at t=3, 12 at t=4, 15 at t=5. To find the solution all that needs to be done is to add these six numbers to get 45 m, but apparently this is wrong. The correct answer is 37.5 m. Why is this?