Let an be the sum of the first $n$ perfect squares, that is, $a_{n} = \displaystyle{}\sum_{k = 1}^{n} k^{2}.$ Show that the sequence $\{a_{n}\}$ satisfies the linear nonhomogeneous recurrence relation $a_{n} = a_{n-1} + n^{2}$ and the initial condition $a_{1} = 1.$ Use
Theorem $6$ to determine a formula for $a_{n}$ by solving this recurrence relation.
