Prove theoretically that the paths traversed by the body during uniformly accelerated rectilinear

Prove theoretically that the paths traversed by the body during uniformly accelerated rectilinear motion are related as a sequential series of odd numbers.

Since the initial velocity is 0, the Coordinate of the body is:
x = at2 / 2
In time 2t the coordinate becomes equal to 2at2: 3t – 4.5at2: 4t – 8at2
During the second period of time, the body has passed 1.5at2;
third – 2.5at2;
fourth – 3.5at2
S4 / S3 / S2 / S1 = 7: 5: 3: 1
As you can see, for equal periods of time, the paths traveled differ as odd numbers.