Let an be the sum of the first $n$ triangular numbers, that is,
$a_{n} = \displaystyle{}\sum_{k = 1}^{n} t_{k},\:\text{where}\: t_{k} = k(k + 1)/2.$ Show that $\{an\}$ satisfies the linear nonhomogeneous recurrence relation $a_{n} = a_{n-1} + n(n + 1)/2$ and the initial condition $a_{1} = 1.$
Use Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.
