Fibonacci and Lucas Identities Derived via the Golden Ratio

Abstract

By expressing Fibonacci and Lucas numbers in terms of the powers of the golden ratio α = (1 + √ 5) / 2 and its inverse β = − 1 /α = (1 − √ 5) / 2 , a multitude of Fibonacci and Lucas identities have been developed in the literature. In this paper, the reverse course is followed: numerous Fibonacci and Lucas identities are derived by making use of the well-known expressions for the powers of α and β in terms of Fibonacci and Lucas numbers.