A Study on Fibonacci Numbers and the Golden Ratio

A. Kiruthiga, V. Manjubashini, Mohana Priya

Fibonacci numbers; golden ratio; Fibonacci sequence; Q-matrix, Binet’s formula

Fibonacci numbers and the golden ratio are fundamental concepts in number theory and mathematics. The Fibonacci sequence is defined by a recursive relation in which each term is the sum of the two preceding terms. The golden ratio is a mathematical constant closely related to the Fibonacci sequence. This paper presents definitions, properties, identities, and important results related to Fibonacci numbers and the golden ratio. Important identities such as the sum of Fibonacci numbers, sum of squares, Fibonacci Q-matrix, Cassini’s identity, and Binet’s formula are discussed. The relationship between Fibonacci numbers and the golden ratio is analyzed. Applications in nature, computer science, and mathematics are also presented.

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