Progressões aritméticas de ordem superior : resultados e aplicações

Abstract

This dissertation is a study on the Arithmetic Progressions of Higher Order and, to show the main results of this work, it was necessary to present the Principle of Mathematical Induction and the Principles of Discrete Mathematics. We begin the work by defining the Principle of Mathematical Induction and using it to demonstrate identities, inequalities and to solve some problems of divisibility. Next, we present the principles of discrete mathematics, the additive principle, and the multiplicative principle. We begin the study of progressions by exhibiting some elementary notions of succession. Promptly, we present the concept of ordinary arithmetic progressions, higher order arithmetic progressions, and some results. As an application, we present a conjecture based on higher order arithmetic progressions and a general formula for calculating the number of subtriangles of a larger triangle with sides equal to n, for every natural n. The reason for presenting the conjecture based on arithmetic progressions is due to the fact that we plan an activity in which we present this concept to high school students and then ask them to look for a possible formula for the number of subtriangles of a larger side triangle with n side points (with odd n).