UM MÉTODO ALTERNATIVO PARA OBTENÇÃO DA FÓRMULA POSICIONAL DE RECORRÊNCIAS LINEARES HOMOGÊNEAS DE QUARTA ORDEM COM COEFICIENTES CONSTANTES

Abstract

Numerical sequences play a central role in mathematics, being fundamental to the modeling of natural phenomena, the analysis of algorithms, and the development of more complex mathematical models. In this context, obtaining analytical solutions—especially through closed-form expressions—is of particular importance, as it allows for the exact computation of sequence terms, thus overcoming the limitations inherent in approximate numerical methods. In light of this, the present article introduces an alternative method for determining the closed-form expression of fourth-order linear homogeneous recurrences with constant coefficients. As an application of the proposed method, a generalization of the Fibonacci sequence is presented, demonstrating both the effectiveness and flexibility of the approach, while also highlighting the relevance of the theoretical study of recurrences in understanding and extending classical mathematical patterns.

Keywords: Homogeneous linear recurrences; positional formula; generalization of the Fibonacci sequence; fourth-order equations.