A Fibonacci and Lucas Inequality

In this presentation, we will outline the process of solving an open problem from the journal The Fibonacci Quarterly. Specifically, we discuss the solution to the following inequality: For n≥4 prove that: 1/(〖〖(F〗_n〗^2-1)(〖F_(n+1)〗^2-1)〖〖(L〗_n〗^2-1))+1/(〖〖(F〗_n〗^2+1)(〖F_(n+1)〗^2+1) 〖〖(L〗_n〗^2+1))>2/〖F_(n+1)〗^6 where F_n and L_n are the nth Fibonacci and Lucas numbers, respectively. We recall that the Fibonacci sequence is defined by the recurrence relation F_n=F_(n-1)+F_(n-2) for n≥2 with initial values F_0=0 and F_1=1. Similarly, that the Lucas sequence is defined by the recurrence relation L_n=L_(n-1)+L_(n-2) for n≥2 with initial values L_0=2 and L_1=1. In this talk, we discuss the essential strategies we used to solve the problem. These include: a heuristic approach, a divide-and-conquer method, and a theoretical proof. Finally, I would like to share our experience working on this problem and what we gained throughout the process.