Question

When acceleration is constant and the initial velocity is zero, displacement is directly proportional to the square of time for which the linear motion has taken place.

Answer 1

Now, $S=\frac{1}{2}g{T}^2$
if you will divide the total time T into N equal intervals the displacements of individual time intervals should add up to total displacement. Now time is increasing in equal units and forming a complete square at each frame seems like I have seen such a pattern.
$1=1^2$
$1+3=2^2$
$1+3+5=3^2$
$1+3+...+(2n-1)=n^2$
from the above pattern, it is evident the dividing the total in N interval will divide the total displacement in 1:3:5:...(2N-1) parts.