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The Fibonacci Sequence and Generalizations Abstract: This paper gives a brief introduction to the famous Fibonacci sequence and demonstrates the close link between matrices and Fibonacci numbers. The much-studied Fibonacci sequence is defined recursively by the equation yk+2 = yk+1 + yk, where y1 = 1 and y2=1. By using algebraic properties of matrices, we derive an explicit formula for the kth Fibonacci number as a function of k and an approximation for the "golden ratio" yk+1 / yk. We also demonstrate how useful eigenvectors and eigenvalues can be in understanding the dynamics of linear recurrence relations of the form yk+2 = ayk+1 + byk where a, b ? R. I. Introduction The Fibonacci sequence, probably one of the oldest and most famous sequences of integers, has fascinated both amateur and professional mathematicians for centuries. Named after its originator, Leonardo Fibonacci, the Fibonacci sequence occurs frequently in nature and has numerous...

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