Problem 1.Consider the ratio between two consecutive Fibonacci numbersfnfor n= 2, 3, ....I'n =fn-1(a)Show that the sequence of ratios (rn)nEN satisfies the recursive relation11+T'n-1T'n(b)Assume that this sequence converges. Show that the limit rị = limn+x rnis a fixed point of the function f(r) = 1+, i.e., f(ri) = r1.(c)Show that ri is precisely the 'golden ratio', r = }(1+ v5).

Name: Problem 1.Consider the ratio between two consecutive Fibonacci number
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Description: Problem 1.Consider the ratio between two consecutive Fibonacci numbersfnfor n= 2, 3, ....I'n =fn-1(a)Show that the sequence of ratios (rn)nEN satisfies the recursive relation11+T'n-1T'n(b)Assume that this sequence converges. Show that the limit rị =

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Consider the ratio between two consecutive Fibonacci numbers fn for n = 2, 3, .... I'n fn-1 (a) Show that the sequence of ratios (rn)nEN satisfies the recursive relation 1 T'n = 1+ T'n-1 Assume that this sequence converges.' Show that the limit ri is a fixed point of the function f(r) = 1+, i.e., f(ri) = r1. Show that r is precisely the 'golden ratio', ry (1+ V5). (b) lim,+0 "'n (c)

Consider the ratio between two consecutive Fibonacci numbers fn for n = 2, 3, .... I'n fn-1 (a) Show that the sequence of ratios (rn)nEN satisfies the recursive relation 1 T'n = 1+ T'n-1 Assume that this sequence converges.' Show that the limit ri is a fixed point of the function f(r) = 1+, i.e., f(ri) = r1. Show that r is precisely the 'golden ratio', ry (1+ V5). (b) lim,+0 "'n (c)

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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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Problem 1.
Consider the ratio between two consecutive Fibonacci numbers
fn
for n
= 2, 3, ....
I'n =
fn-1
(a)
Show that the sequence of ratios (rn)nEN satisfies the recursive relation
1
1+
T'n-1
T'n
(b)
Assume that this sequence converges. Show that the limit rị = limn+x rn
is a fixed point of the function f(r) = 1+, i.e., f(ri) = r1.
(c)
Show that ri is precisely the 'golden ratio', r = }(1+ v5).

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