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: $$\sum_{k=1}^n a_kb_k = A_nb_{n+1}+\sum_{k=1}^n A_k(b_k-b_{k+1})$$ with $A_k:=\sum_{i=1}^k a_i.$
Proofs: 1