Contextualization

Geometric Progression (G.P.) is a numerical sequence where each term is the result of multiplying the previous term by a constant (ratio). It is an important mathematical tool that appears in many real-world contexts, where a multiplicative factor is constant.

In mathematics, a geometric progression is a numerical sequence where each term after the first is found by multiplying the previous term by a fixed constant, called the ratio. If the ratio is greater than 1, we obtain an increasing sequence (2, 4, 8, 16,...), while if the ratio is less than 1, we obtain a decreasing sequence (10, 5, 2.5, 1.25,...).

In number theory, geometric progressions are used to study prime and composite numbers, as well as to analyze the nature of irrational and transcendental numbers. Geometric progressions are also used in analysis to study exponential and logarithmic functions, and in calculus to derive and integrate these functions. Additionally, geometric progressions are important tools in physics, biology, economics, demography, and many other areas of science and engineering.

Application in the Real World

Geometric progressions can model many real-world phenomena. For example, compound interest in economics is a classic example of a geometric progression. An amount borrowed at a fixed interest rate will grow according to a geometric progression.

Another example is the reproduction of microorganisms. Assuming each cell divides into two each day, the number of cells will form a geometric progression. Geometric progressions are also used in computer science in the analysis of algorithm complexity, where often the amount of time or space required to solve a problem increases geometrically with the size of the problem.

Therefore, the study of geometric progressions is fundamental, not only for the development of mathematical skills, but also for understanding and solving real-world problems.

Atividade Prática

Activity Title: "Exploring the World through Geometric Progressions"

Project Objective

This project aims to research and create a document on practical everyday situations where geometric progressions (G.P.) can be found. In addition, students must calculate the terms of each found G.P., thus demonstrating an understanding of the concept and the application of its formulas.

Detailed Project Description

Students, divided into groups of 3 to 5, should research and identify, in everyday situations, where geometric progressions occur. At least 5 examples will be required. Situations such as population growth, appreciation or depreciation of an asset, cell growth, among others, can be considered.

Additionally, they must calculate the 10th term of each found G.P., demonstrating an understanding of the concept and the application of its formulas.

Finally, the groups must compile all the information into a final report, explaining each example and its calculations.

Necessary Materials

For the project, the following materials will be needed:

• Computer with internet access for research.
• Mathematics textbook and/or support materials provided by the teacher.
• Writing materials (paper and pen) or word processors (Microsoft Word, Google Docs).

Detailed Step-by-Step for Activity Completion

1. Research: Students will conduct research in groups, either online or in books, to identify practical everyday situations where geometric progressions occur.
2. Examples: Each group must choose 5 different situations where geometric progressions occur.
3. Calculations: For each chosen situation, the group must calculate the 10th term of the G.P., showing the step-by-step calculation.
4. Documentation: Students must compile the obtained information (research, discussion, calculations) into a report.
5. Review: The entire group must review the report before submission, correcting any errors and ensuring the clarity and accuracy of the information.

Project Deliverables

The final product will be a written report containing:

• Introduction: Description of what a geometric progression is and explanation of its relevance and application in the real world, as well as the project's objective.
• Development: Presentation of the examples found, explaining the situation, showing the G.P., and detailing the calculations made.
• Conclusions: Final reflections on what was learned, the challenges encountered and how they were overcome, and the conclusions drawn from the practice.
• Bibliography: List of sources used for the project.

This project requires time management and teamwork, as students will have to divide tasks, collaborate, communicate effectively, and solve problems together. Additionally, they will be encouraged to think creatively in search of everyday situations involving geometric progressions.