The following summation method can be used for sums of products and is due to Niels Henrik Abel (1802 - 1829).
Let a_1,\ldots,a_n and b_1,\ldots,b_n,b_{n+1} be some given elements of a unit ring (R,+,\cdot). Then it is possible to reformulate the sum \sum_{k=1}^n a_kb_k as follows: \sum_{k=1}^n a_kb_k = A_nb_{n+1}+\sum_{k=1}^n A_k(b_k-b_{k+1})
Proofs: 1