Galileo's law of odd numbers: "The distances traversed, during equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity [namely, $1: 3: 5: 7 \ldots$]". Prove it.

(N/A) Let us divide the time interval of motion of an object under free fall into many equal intervals $\tau$ and find out the distances traversed during successive intervals of time. Since the initial velocity is zero, the position $y$ at time $t$ is given by:
$y = -\frac{1}{2} g t^2$
Using this equation, we calculate the position of the object after time intervals $t = 0, \tau, 2\tau, 3\tau, \dots$. If we define $y_0 = -\frac{1}{2} g \tau^2$ as the position after the first interval $\tau$, then the position at time $n\tau$ is $n^2 y_0$. The distance traversed in the $n^{th}$ interval is the difference between the position at $n\tau$ and $(n-1)\tau$:
Distance in $n^{th}$ interval $= |n^2 y_0 - (n-1)^2 y_0| = |(n^2 - (n^2 - 2n + 1)) y_0| = (2n - 1) |y_0|$.
For $n = 1, 2, 3, 4, \dots$, the distances are $1|y_0|, 3|y_0|, 5|y_0|, 7|y_0|, \dots$. Thus, the ratio of distances is $1: 3: 5: 7: \dots$, which are odd numbers. This law was established by Galileo Galilei ($1564$-$1642$), who was the first to make quantitative studies of free fall.