Some miscellaneous results of the Fibonacci sequence and the golden ratio

Document Type : Research Articles

Authors

1 Department of Mathematics, Yasouj University, Yasouj, Iran

2 ‎‎Department of Applied Mathematics‎, ‎Faculty of Mathematics and Computer‎, ‎Shahid Bahonar University of Kerman, Kerman‎, ‎Iran.

3 Farhangian University, Yasouj‎, ‎Iran

4 ‎‎Department of Mathematics‎, ‎Vali-e-Asr University of Rafsanjan, Rafsanjan‎, ‎Iran

10.22080/cjms.2025.29210.1759

Abstract

The Fibonacci sequence and the golden ratio for centuries due to their deep mathematical properties and diverse applications in theoretical and applied fields. This paper explores the mathematical relationships between these two concepts and their practical uses in different fields. We construct a power series using Fibonacci numbers and demonstrate that the radius of convergence of this series is equal to the golden ratio. Furthermore, we investigate the conditions under which three Fibonacci numbers can form a triangle and analyze the properties of such triangles. We also introduce the concept of pseudo-right-angled triangles and provide a characterization of these figures. Finally, we analyze and decompose the polynomial \( x^n - F_n x - F_{n-1} = 0 \), a relation in which the golden ratio emerges as a root.

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