Pardon the cryptic notation and possibly trivial question. I believe the following holds.
Define $X_t=(\prod_{i\leq t-1}a_i)(\prod_{j\geq t+1} b_j)(a_t-b_t).$ Show that $\prod_{i=1}^na_i-\prod_{j=1}^nb_i=\sum_{t=1}^{n-1}X_t.$
Is there a quick and elegant way of seeing this? It's straightforward by induction and I think as well by multilinearity of determinants and possibly even by inclusion-exclusion. Moreover, it should be also possible from expanding $(a_1-b_1)(a_2-b_2)...(a_{n-1}-b_{n-1})(a_n+(-1)^nb_n)$.