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Prove these properties on Fibonacci and Lucas Sequences

Source: 2019 Jozsef Wildt International Math Competition-W. 19

May 18, 2020
number theoryFibonacci sequenceLucas sequence

Problem Statement

Let {Fn}n∈Z\{F_n\}_{n\in\mathbb{Z}} and {Ln}n∈Z\{L_n\}_{n\in\mathbb{Z}} denote the Fibonacci and Lucas numbers, respectively, given by Fn+1=Fn+Fn−1 and Ln+1=Ln+Ln−1 for all n≥1F_{n+1} = F_n + F_{n-1}\ \text{and}\ L_{n+1} = L_n + L_{n-1}\ \text{for all}\ n \geq 1with F0=0F_0 = 0, F1=1F_1 = 1, L0=2L_0 = 2, and L1=1L_1 = 1. Prove that for integers n≥1n \geq 1 and j≥0j \geq 0
[*]∑k=1nFk±jLk∓j=F2n+1−1+{0,if n is even(−1)±jF±2j,if n is odd\sum \limits_{k=1}^n F_{k\pm j}L_{k\mp j}=F_{2n+1}-1+\begin{cases} 0, & \text{if}\ n\ \text{is even}\\ \left(-1\right)^{\pm j}F_{\pm 2j}, & \text{if}\ n\ \text{is odd} \end{cases} [*] ∑k=1nFk+jFk−jLk+jLk−j=F4n+2−1−nL4j5\sum \limits_{k=1}^nF_{k+j}F_{k-j}L_{k+j}L_{k-j}=\frac{F_{4n+2}-1-nL_{4j}}{5}
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