Closed forms for sums of integers and their squares

Express $1+2+\cdots+n$ and $1^2+2^2+\cdots+n^2$ as closed-form functions of $n$.

$\tfrac{n(n+1)}{2}$ and $\tfrac{n(n+1)(2n+1)}{6}$. The first follows by pairing $k$ with $n+1-k$. For the second, telescope $(k+1)^3-k^3=3k^2+3k+1$ over $k=1,\dots,n$ and solve for $\sum k^2$ (or verify by induction).