A Fibonacci and Lucas Inequality
School Name
Academic Magnet High School
Grade Level
11th Grade
Presentation Topic
Mathematics
Presentation Type
Mentored
Abstract
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.
Recommended Citation
Richards, Nora, "A Fibonacci and Lucas Inequality" (2025). South Carolina Junior Academy of Science. 122.
https://scholarexchange.furman.edu/scjas/2025/all/122
Location
PENNY 203
Start Date
4-5-2025 9:15 AM
Presentation Format
Oral Only
Group Project
No
A Fibonacci and Lucas Inequality
PENNY 203
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.
