kinematics - Galileo's law of odd numbers

The Galileo’s law of odd numbers states that the distances traveled are proportional to the squares of the elapsed times. In other words, in equal successive periods of time, the distances traveled by a free-falling body are proportional to the succession of odd numbers($1, 3, 5, 7,$ etc.).

I clearly understand from kinematics equation that the distances traversed in a time interval are $-\frac{1}{2} gt^2$ so it will be proportional to squares of elapsed times. But what I don't get is how is it proportional to the succession of odd numbers?

Answer

This is because $(n+1)^2-n^2=2n+1$, which is odd. Hence, as in the 1st second, $d_1=-\frac{g} {2} $, in the 2nd second, $d_2=-\frac{4g} {2} $, and so on, we get $\frac{d_2 - d_1}{d_1}=\frac{4-1}{1}=3=2.1+1$