# Fibonacci Numbers and Their Lucas Coefficients

## TL;DR

This paper proves a new identity linking Fibonacci and Lucas numbers, expressing Fibonacci numbers as a sum involving Lucas coefficients, with a detailed inductive proof and numeric illustration.

## Contribution

It introduces a novel identity connecting Fibonacci and Lucas sequences, providing a detailed proof and practical example.

## Key findings

- Established a new Fibonacci-Lucas identity for all n >= 2
- Provided a rigorous inductive proof of the identity
- Illustrated the identity with a numerical example

## Abstract

We show that for the classical Fibonacci sequence (Fn) and the Lucas sequence (Ln) the following identity holds for every integer n >= 2: (n-1)Fn equals the sum from k=1 to n-1 of Lk multiplied by F(n-k). Equivalently, this gives a representation of the nth Fibonacci number as Fn = (1 / (n-1)) times the same sum. We present a detailed proof by mathematical induction and illustrate the identity with a numeric example.