The Fibonacci sequence 1, 1, 2, 3, 5, ... is the sequence (fn)nɛN defined by fi = 1, f2 = 1, and fn = fn-1+ fn-2 if n 2 3. The aim of this question is to prove that the sequence (an) of rational numbers an = converges to a limit. (The limit is called the golden ratio.) fn+1/fn (a) Show that the sequence (an) satisfies 1 if n > 2. a1 = 1, an =1+ ап-1 (b) Sketch a graph (as done in lectures) of the sequence (an) for 1 a2n+1•
The Fibonacci sequence 1, 1, 2, 3, 5, ... is the sequence (fn)nɛN defined by fi = 1, f2 = 1, and fn = fn-1+ fn-2 if n 2 3. The aim of this question is to prove that the sequence (an) of rational numbers an = converges to a limit. (The limit is called the golden ratio.) fn+1/fn (a) Show that the sequence (an) satisfies 1 if n > 2. a1 = 1, an =1+ ап-1 (b) Sketch a graph (as done in lectures) of the sequence (an) for 1 a2n+1•
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:2. The Fibonacci sequence 1,1, 2, 3, 5, ... is the sequence (fn)nƐN defined by
fi = 1, f2 = 1, and fn = fn-1 + fn-2 if n > 3.
The aim of this question is to prove that the sequence (an) of rational numbers an
converges to a limit. (The limit is called the golden ratio.)
fn+1/fn
(a) Show that the sequence (an) satisfies
1
if n > 2.
a1 = 1, an =1+
An-1
(b) Sketch a graph (as done in lectures) of the sequence (an) for 1<n< 5.
(c) Prove that for all n E N,
a2n-1 < a2n+1 < a2n and a2n > a2n+2 > a2n+1•
(d) Prove that the subsequences (a2n-1)nEN and (a2n)nEN are bounded monotone sequences.
(e) Prove that (a2n-1)neN Converges to a limit L that satisfies the equation
1
L = 1+
1++
(Hint: compare the convergence of (a2n-1)nEN to (a2n+1)nƐN.)
(f) Find L. (Solve the equation.)
(g) Prove that the subsequence (a2n)nɛN also converges to L.
(h) Finally, prove that (an)nEN Converges to L.
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