Contextualization

Mathematics is a powerful tool that not only allows us to understand how the world works, but also to predict the behavior of various phenomena and situations. Within mathematics, one of the most important areas of study is the analysis of sequences and series, particularly Geometric Progressions (GPs). GPs are numerical sequences where each subsequent term is obtained by multiplying the previous term by a constant.

The concept of Geometric Progression is fundamental to various fields of knowledge, such as physics, economics, biology, and engineering. For example, in the study of compound interest in economics, the formula for calculating the future value of an investment is a GP. In physics, many natural phenomena, such as the free fall of an object, can be modeled using GPs. Furthermore, the notion of limit, a key idea in calculus, is based on the sum of infinite series, which is an extension of the concept of summing GPs.

In this project, we will explore the sum of a Geometric Progression (GP), which is one of the fundamental operations that can be performed on a GP. The sum of the terms of a GP is useful in various practical situations, such as calculating the sum of accumulated interest on an investment, modeling the spread of an epidemic, or predicting the total production of a factory.

The ability to calculate the sum of a GP is not only an important milestone in mathematical education, but also a powerful tool that opens the door to understanding and solving complex real-life problems.

Below are some reliable sources that can be used as references to deepen the understanding of the concept of geometric progression and its sum:

1. Book: "Fundamentos de Matemática Elementar" - Vol. 3, by Gelson Iezzi and Carlos Murakami.
2. Website: Khan Academy (Geometric Progressions: https://pt.khanacademy.org/math/precalculus/x9e81a4f98389efdf:series/x9e81a4f98389efdf:geometric-series/a/intro-geometric-series).
3. Video: "What are Arithmetic and Geometric Progressions?" from Matemática Rio Channel with Prof. Rafael Procópio (https://www.youtube.com/watch?v=KK-8aRgFxqo).

Practical Activity

Activity Title:

"Exploring the world of Geometric Progressions: An analysis of numerical sequences in economics and engineering"

Project Objective:

This project aims to offer students the opportunity to deepen their knowledge and understanding of the concept and application of the sum of a Geometric Progression (GP). For this purpose, practical approaches related to real-life situations in the fields of economics and engineering will be used.

Detailed Project Description:

The student groups (3 to 5 students per group) should choose 2 real problems, one in the field of economics and another in the field of engineering, where the concept of the sum of a GP is applied. After choosing the problems, they should develop a detailed study of how the sum of the GP is applied in these contexts.

Students will have the opportunity to delve into the domains of economics and engineering and apply the knowledge acquired in mathematics to real situations, developing research skills, teamwork, problem-solving, and communication.

Required Materials:

• Reference books on the topic (to be used for researching the application of the sum of a GP in real situations)
• Computer with internet access (for research and for the preparation of the final report)
• Text editing software (for the preparation of the final report)

Detailed Step-by-Step for the Activity:

1. Research of real problems (6 hours): Students should research real situations in economics and engineering where the concept of the sum of a GP is applied. Each team must choose at least 2 problems, one in the field of economics and another in the field of engineering.

2. Study of the problems (6 hours): Each team should conduct an in-depth study on how the sum of the GP is applied in the chosen problems.

3. Problem-solving (6 hours): Applying the concept of the sum of the GP, students should find a solution to the chosen problems.

4. Final Report (6 hours): Each group must produce a final report containing a detailed description of the chosen problems, the application of the sum of the GP to solve them, and the results obtained.

Project Deliverables:

The final project deliverable will be a report, in the format of a scientific article, written by the students containing the following chapters:

1. Introduction: Should contain the presentation of the chosen problems, the contextualization of these problems, the justification for their choice, as well as explaining the objectives of the work.

2. Development: Should present the study conducted by the students, the theory of the sum of a GP applied in solving the problems, and the methodology used for this. The description of the team's work process should also be included.

3. Results and Discussion: Should present the solution found for the problems and a discussion of the results obtained. It is important to connect, at this point, the theory learned with the practice carried out, demonstrating the learning of the studied concepts.

4. Conclusion: Should address the team's learnings during the project, the difficulties faced, and the solutions found to overcome them. A reflection on the relevance of mathematics in solving real problems should also be made.

5. Bibliographical References: References for all bibliographic and webgraphic material used during the research and preparation of the work should be cited at this point.

In the report, students should clearly explain the process carried out, the results obtained, and the conclusions. The final report should have an average of 20 pages and be delivered in digital format to the teacher.