Three visual approaches to binomial Fibonacci identities

We present three visual proofs of the classical identity ∑k=0n(nk)Fk=F2n, first discovered by Édouard Lucas. Each proof exploits the combinatorial structure of Pascal’s triangle, or a variant thereof, to uncover the link between binomial coefficients and Fibonacci numbers using a technique called arrow-chasing. All three approaches naturally extend to general second-order and certain higher-order recurrence sequences, such as the Jacobsthal and Padovan numbers, providing unified visual arguments for a broad family of binomial identities.