Paper 4, Section I, 2E2 E

Numbers and Sets | Part IA, 2021

The Fibonacci numbers FnF_{n} are defined by F1=1,F2=1,Fn+2=Fn+1+Fn(n1)F_{1}=1, F_{2}=1, F_{n+2}=F_{n+1}+F_{n}(n \geqslant 1). Let an=Fn+1/Fna_{n}=F_{n+1} / F_{n} be the ratio of successive Fibonacci numbers.

(i) Show that an+1=1+1/ana_{n+1}=1+1 / a_{n}. Hence prove by induction that

(1)nan+2(1)nan(-1)^{n} a_{n+2} \leqslant(-1)^{n} a_{n}

for all n1n \geqslant 1. Deduce that the sequence a2na_{2 n} is monotonically decreasing.

(ii) Prove that

Fn+2FnFn+12=(1)n+1F_{n+2} F_{n}-F_{n+1}^{2}=(-1)^{n+1}

for all n1n \geqslant 1. Hence show that an+1an0a_{n+1}-a_{n} \rightarrow 0 as nn \rightarrow \infty.

(iii) Explain without detailed justification why the sequence ana_{n} has a limit.

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