The four essential formulas: sum(k)=n(n+1)/2, sum(k^2)=n(n+1)(2n+1)/6, sum(k^3)=[n(n+1)/2]^2, sum(k^4)=n(n+1)(2n+1)(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)(n+2)/3, sum(k(k+1)(k+2))=n(n+1)(n+2)(n+3)/4 (pattern continues). These formulas allow evaluation of any polynomial sum: expand the polynomial, sum each power separately. Example: sum(3k^2+2k+1)=3sum(k^2)+2sum(k)+n.