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Journal of Mathematics
Volume 2013, Article ID 204674, 3 pages
http://dx.doi.org/10.1155/2013/204674
Research Article

From Fibonacci Sequence to the Golden Ratio

1Dipartimento di Architettura, Università di Napoli, Via Monteoliveto 3, 80134 Napoli, Italy
2Istituto per le Applicazioni del Calcolo “Mauro Picone”, Sezione di Napoli, Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111, 80131 Napoli, Italy
3Dipartimento di Matematica, Università di Salerno, Via Ponte Don Melillo 4, Fisciano, 84084 Salerno, Italy

Received 14 November 2012; Accepted 11 February 2013

Academic Editor: Mike Tsionas

Copyright © 2013 Alberto Fiorenza and Giovanni Vincenzi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the well-known characterization of the Golden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence, and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratio coincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but, from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots, which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of the characterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, without any assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.