Let \((a_k)_{k\ge1}\) and \((b_k)_{k\ge1}\) be sequences of complex numbers, and put \(A_n \coloneqq \sum_{k=1}^n a_k\) and \(B_n\coloneqq\sum_{k=1}^n b_k\) for \(n\ge1\), with \(A_0 \coloneqq 0\) and \(B_0\coloneqq0\).
For all integers \(1\le m\le n\), \[ \sum_{k=m}^n a_k b_k = A_nb_n - A_{m-1} b_m - \sum_{k=m}^{n-1} A_k(b_{k+1} - b_{k}). \]
For all sequences \((f_k)\) and \((g_k)\) of complex numbers and all integers \(m\le n\), writing \(\Delta g_k \coloneqq g_{k+1} - g_k\) and \(\Delta f_k\coloneqq f_{k+1}-f_k\), \[ \sum_{k=m}^n f_k \, \Delta g_k = f_{n+1} g_{n+1} - f_m g_m - \sum_{k=m}^n g_{k+1} \, \Delta f_k. \]
For every \(n\ge1\), \[ A_{n} B_{n}=\sum_{k=1}^{n} A_{k} b_{k}+\sum_{k=1}^{n} a_{k} B_{k-1}. \]
For (b), \(f_{k+1}g_{k+1}-f_kg_k=f_k\,\Delta g_k+g_{k+1}\,\Delta f_k\) for every \(k\); summing over \(m\le k\le n\) telescopes the left side to \(f_{n+1}g_{n+1}-f_mg_m\).
For (a), substitute \(a_k=A_k-A_{k-1}\): \[ \sum_{k=m}^n a_kb_k=\sum_{k=m}^nA_kb_k-\sum_{k=m-1}^{n-1}A_kb_{k+1}=A_nb_n-A_{m-1}b_m-\sum_{k=m}^{n-1}A_k(b_{k+1}-b_k). \]
For (c), \(A_kB_k-A_{k-1}B_{k-1}=A_k(B_{k-1}+b_k)-(A_k-a_k)B_{k-1}=A_kb_k+a_kB_{k-1}\) for every \(k\ge1\); summing over \(1\le k\le n\) and using \(A_0B_0=0\) gives the identity.
Identity (a) is the discrete analogue of integration by parts: for \(F\) an antiderivative of \(f\) on \([a,b]\) and \(g\) continuously differentiable, \[ \int_a^b f g = F(b)g(b) - F(a)g(a) - \int_a^b Fg'. \] The partial sums \(A_k\) play the role of \(F\), and the differences \(b_{k+1}-b_k\) the role of \(g'\).