From Fibonacci Sequence to the Golden Ratio
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 i...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/204674 |
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| author | Alberto Fiorenza Giovanni Vincenzi |
| author_facet | Alberto Fiorenza Giovanni Vincenzi |
| author_sort | Alberto Fiorenza |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-3863b6ba4194447c8c9eb7da00584459 |
| institution | Kabale University |
| issn | 2314-4629 2314-4785 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Mathematics |
| spelling | doaj-art-3863b6ba4194447c8c9eb7da005844592025-02-03T01:11:47ZengWileyJournal of Mathematics2314-46292314-47852013-01-01201310.1155/2013/204674204674From Fibonacci Sequence to the Golden RatioAlberto Fiorenza0Giovanni Vincenzi1Dipartimento di Architettura, Università di Napoli, Via Monteoliveto 3, 80134 Napoli, ItalyDipartimento di Matematica, Università di Salerno, Via Ponte Don Melillo 4, Fisciano, 84084 Salerno, ItalyWe 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.http://dx.doi.org/10.1155/2013/204674 |
| spellingShingle | Alberto Fiorenza Giovanni Vincenzi From Fibonacci Sequence to the Golden Ratio Journal of Mathematics |
| title | From Fibonacci Sequence to the Golden Ratio |
| title_full | From Fibonacci Sequence to the Golden Ratio |
| title_fullStr | From Fibonacci Sequence to the Golden Ratio |
| title_full_unstemmed | From Fibonacci Sequence to the Golden Ratio |
| title_short | From Fibonacci Sequence to the Golden Ratio |
| title_sort | from fibonacci sequence to the golden ratio |
| url | http://dx.doi.org/10.1155/2013/204674 |
| work_keys_str_mv | AT albertofiorenza fromfibonaccisequencetothegoldenratio AT giovannivincenzi fromfibonaccisequencetothegoldenratio |