Sum of two consecutive triangular numbers

Statement

The sum of two consecutive triangular numbers is a square: $T_{k-1}+T_k=k^2$. Compute $T_{15}+T_{16}$ and identify which square it is.

Solution

With $k=16$ we get $T_{15}+T_{16}=k^2=16^2$. Direct check: $T_{15}=\frac{15\cdot 16}{2}=120,\qquad T_{16}=\frac{16\cdot 17}{2}=136,$ $T_{15}+T_{16}=120+136=\boxed{256}=\boxed{16}^2.$