The 99th Fibonacci Identity

In the book Proofs that Really Count [1], the authors use combinatorial arguments to prove many identities involving Fibonacci numbers, Lucas numbers, and their generalizations. Among these, they derive 91 of the 118 identities mentioned in Vajda’s book [2], leaving 27 identities unaccounted. Eight of these identities, presented later in this paper, have such a similar appearance, the authors remark (on page 144) that “one good idea might solve them all.” In this paper, we provide elegant combinatorial proofs of these Fibonacci and Lucas identities along with generalizations to arbitrary initial conditions. Before examining these new identities, we warm up with the following well known identity, which will allow us to define terminology and illustrate our approach. Identity 1. For n ≥ 0,