Golden Ratio and Sums of Ratios in Fibonacci Numbers

Abstract:

We solved an open problem from the August 2022 edition of The Fibonacci Quarterly journal. We proved that a sum and a product are ratios of consecutive integers and consecutive Fibonacci numbers respectively. We then further evaluated an infinite product involving Fibonacci numbers. This proof was completed using Fibonacci and Lucas identities, the golden ratio, mathematical induction, and properties of summations and products.