Lagrange Interpolation Formula (LIF) has an important role in problems about unknown polynomials with known values in only some points. However this is not the only critic role of LIF. In this paper we shed light on some less known aspects of LIF. We use it to prove identities, and we show its relation with derivation and its applications to arithmetic progressions of higher order, introducing the finite dierences operator which opens a fan of possibilities for students preparing for mathematical competitions.