CMOQR 2019 Problem 2

Rosemonde is stacking spheres to make pyramids. She constructs two types of pyramids $S_n$ and $T_n$ . The pyramid $S_n$ has $n$ layers, where the top layer is a single sphere and the $i^{th}$ layer is an $i\times$ i square grid of spheres for each $2 \le i \le n$ . Similarly, the pyramid $T_n$ has $n$ layers where the top layer is a single sphere and the $i^{th}$ layer is $\frac{i(i+1)}{2}$ spheres arranged into an equilateral triangle for each $2 \le i \le n$ .