Show that, for any polynomial $P$ of degree at most $n - 2$, $$\sum _ { i = 1 } ^ { n } \frac { P \left( a _ { i } \right) } { \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \left( a _ { i } - a _ { j } \right) } = 0 .$$
Show that, for any polynomial $P$ of degree at most $n - 2$,
$$\sum _ { i = 1 } ^ { n } \frac { P \left( a _ { i } \right) } { \prod _ { \substack { j = 1 \\ j \neq i } } ^ { n } \left( a _ { i } - a _ { j } \right) } = 0 .$$