Relações de recorrências lineares de primeira ordem: uma proposta para modelar problemas no ensino médio através de fórmulas recursivas

Abstract

This work focuses on the study of numerical sequences, with particular emphasis on first-order linear recurrence relations. We explore how these relations can be applied in various mathematical areas, including geometry. The main goal is to enable students to make meaningful connections between distinct mathematical concepts, fostering a deeper understanding of sequence theory. The effectiveness of this approach is evaluated through two tests, one administered before and one after the intervention. These tests demonstrate an improvement in students’ performance regarding their understanding of numerical sequences. The applications discussed cover both elementary and secondary education, with specific adaptations for each educational level. The primary focus is on developing recursive reasoning and transitioning from recursive formulas to closed formulas, allowing for the precise and efficient calculation of any subsequent term in the sequence. We began our investigation by modeling a specific problem, starting from initial information established as the foundation. Subsequently, we adopted a formal approach grounded in rigorous methods to demonstrate relevant properties and construct the necessary formalizations and proofs. To illustrate the practical application of the developed knowledge, we explored the resolution of three distinct problems: the Tower of Hanoi, the problem of coins arranged in regular hexagons, and a challenge involving colored liquids organized in bottles. Each of these examples concretely demonstrates the versatility and depth of the techniques addressed. It is important to note that existing literature often presents only superficial outlines for deriving recursive formulas, especially in the context of numerical and non-numerical sequences. The focus is frequently on the final result without delving into the underlying steps and fundamental ideas. This study aims to offer a comprehensive understanding of the process of deriving recursive formulas and how to obtain closed formulas from them. Throughout the work, we demonstrate how to transition from specific problems to numerical sequences and closed formulas, providing a detailed and systematic view of this process.