Problem
ALG-B1-M05-P007 A Fibonacci-Type Sequence
#7
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Let \(F_1=F_2=1\), \(F_{n+2}=F_{n+1}+F_n\). Prove that \(F_{n+3}\ge2F_n\).
Express \(F_{n+3}\) through \(F_{n+1}\) and \(F_n\).
\(F_{n+3}=F_{n+2}+F_{n+1}=F_n+2F_{n+1}\). Since \(F_{n+1}\ge F_n\) for \(n\ge1\), \(F_{n+3}\ge3F_n\), hence \(F_{n+3}\ge2F_n\).
A simple estimate through the recurrence.