MathDB
basic Fibonacci sequence properties

Source: 2022 Viet Nam math olympiad for high school students D2 P1

March 21, 2023
algebra

Problem Statement

Given Fibonacci sequence (Fn)(F_n) a) Prove that: for all u,v∈N,u≥1u,v\in \mathbb{N}, u\ge 1, we have:Fu+v=Fu−1Fv+FuFv+1.F_{u+v}=F_{u-1}F_{v}+F_{u}F_{v+1}. b) Prove that: for all n∈N,n≥1n\in \mathbb{N}, n\ge 1, we have:F2n=Fn(Fn−1+Fn+1),F_{2n}=F_n(F_{n-1}+F_{n+1}),F2n+1=Fn2+Fn+12.F_{2n+1}=F_n^2+F_{n+1}^2.
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