Doron Zeilberger has beat the drum rather enthusiastically for this style of argument; see for example his paper The C-finite Ansatz. M. D. Hirschhorn has a paper entitled A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan in which he says,

In this paper I show that in order to prove Ramanujan's statement it is sufficient to check just the first seven cases, and then I do so.

More controversially, Zeilberger goes further and champions the notion of a "semi-rigorous proof," saying that even if we don't have a fully rigorous proof that the finitely many cases we have checked imply the general case, often the finitely many cases are "good enough." However, even if you don't agree with Zeilberger's attitude toward semi-rigorous proofs, he and his collaborators have many examples of fully rigorous proofs of the type you are looking for.