Analogues of a Fibonacci-Lucas Identity

Sury’s 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn’s 2003 book: {\em Proofs that count: The art of combinatorial proof}) has excited a lot of comment. We give an alternate, telescoping, proof of this—and associated—identities and generalize them. We also give analogous identities for other sequences that satisfy a three-term recurrence relation.