Proof of a Fibonacci Identity with Square Root

A second order recurrence relation Fn is called Fibonacci sequence if it satisfies that F0=0, F1=1, and Fn= Fn-1+Fn-2 for n ≥3. Fibonacci used this sequence to model the breeding of a population of rabbits. Fibonacci numbers are also connected to many other different topics of pure mathematics, applied mathematics, art, and economy including number theory, binomial coefficients, Pascal’s triangle, Wall Street stock market, and the Milky Way. In this presentation we show the solution of a new identity related to Fibonacci numbers. This was an open problem that appeared in The Fibonacci Quarterly as B-1305. Our original solution was submitted for publication to the same journal and is under consideration