The four essential formulas: sum(k)=n$\frac{n+1}{2}$, sum($k^2$)=n(n+1)$\frac{2n+1}{6}$, sum($k^3$)=[n$\frac{n+1}{2}$]^2, sum($k^4$)=n(n+1)(2n+1)$\frac{3n^2+3n-1}{30}$. The Nicomachus identity sum($k^3$)=[sum(k)]^2 is the most elegant and most tested. For sums of products: sum(k(k+1))=n(n+1)$\frac{n+2}{3}$, sum(k(k+1)(k+2))=n(n+1)(n+2)$\frac{n+3}{4}$ (pattern continues). These formulas allow evaluation of any polynomial sum: expand the polynomial, sum each power separately. Example: sum(3$k^{2+2k+1}$)=3sum($k^2$)+2sum(k)+n.