Sum of the first triangular numbers

Statement

Compute the sum of the first six triangular numbers $T_1+T_2+\dots+T_6$ and check it is a tetrahedral number.

Solution

The first six triangular numbers are $1,3,6,10,15,21$. Their sum: $1+3+6+10+15+21=\boxed{56}.$ This is $\mathrm{Tet}_6=\dfrac{6\cdot 7\cdot 8}{6}=56$: the sum of the first $n$ triangular numbers is always $\mathrm{Tet}_n=\dfrac{n(n+1)(n+2)}{6}$.