Introduction

Relevance of the Topic

Geometric Progressions (GPs) are everywhere in our daily lives. From population growth, to the appreciation of material goods, to the evolution of financial investments, GPs are a fundamental aspect of mathematics that helps us understand and predict how things change over time.

Contextualization

The Sum of a Geometric Progression (GP) is a tool that allows us to calculate the total amount that something or someone will reach in a given period. This skill is essential not only for the discipline of mathematics, but also for other areas of study and for everyday life. It is the basis for many more advanced mathematical formulas and concepts, and is a prerequisite for understanding and effectively using second-degree, exponential, and logarithmic equations.

Theoretical Development

Components

• General Term (an): It is one of the most fundamental elements of Geometric Progression. 'an' represents any term of the sequence and is calculated through the formula an = a1 * r^(n-1), where a1 is the first term, r is the progression ratio, and n is the term position.

• Progression Ratio (r): Defines the constant growth or decrease rate between the terms of the GP. If the ratio is greater than 1, the progression will be increasing, if it is between 0 and 1, it will be decreasing.

• Sum of the Terms of a Finite GP (Sn): Provides the total of a sequence of terms of a GP up to a certain point. It is calculated using the formula Sn = (a1 * (r^n - 1))/(r-1), where a1 is the first term, r is the progression ratio, and n is the number of terms.

Key Terms

• Geometric Progression (GP): It is a numerical sequence where each term, starting from the second, is equal to the previous term multiplied by a constant called the ratio.

• General Term (an): As explained earlier, it is one of the main elements of a GP and represents any term of the sequence.

• Progression Ratio (r): As discussed above, it is the constant by which each subsequent term is multiplied to obtain the next term.

• Sum (Sn): The sum of a finite GP up to the nth term. It is a vital component for calculating totals, accumulations, and series.

Examples and Cases

• Example 1 - Increasing GP: Given the GP (2, 4, 8, 16, 32), we can calculate the sum of the first 4 terms (n = 4). Using the formula for the sum of a finite GP, we have: Sn = (2 * (2^4 - 1))/ (2-1) = 30. Therefore, the sum of the first 4 terms of this GP is 30.

• Example 2 - Decreasing GP: Let's consider the GP (32, 16, 8, 4, 2) where the first term is greater than the second. To determine the sum of the first 3 terms (n = 3), the formula Sn = (32 * (1 - 2^3))/(1 - 2) can be applied. The result, in this case, is Sn = -52. Although the result is negative, it is important to note that we are taking the sum of a decreasing GP.

• Example 3 - Constant GP: If the ratio r of a GP is 1, all terms will be equal, and therefore, the sum of the first n terms (Sn) will be n times the value of any term a.

In summary, the Sum of a Geometric Progression represents the total amount that a sequence generates at a given moment, which is essential for a deeper understanding of many mathematical phenomena and calculations.

Detailed Summary

Relevant Points

• Understanding of geometric progression (GP): A GP is a numerical sequence in which each term, except the first, is obtained by multiplying the previous term by a constant called the ratio. This definition is fundamental to understand what a GP is and how it behaves.

• Calculation of the general term (an): The general term is any term of a GP. It is calculated through the formula an = a1 * r^(n-1), where a1 is the first term of the GP, r is the GP ratio, and n is the term position. Understanding how to obtain the general term is crucial for calculating sums of GPs.

• Identification of the ratio (r): The ratio is the number by which each term of the GP is multiplied to obtain the next term. The ratio is a constant and is extremely important in the definition and calculation of a GP.

• Formula for the Sum of the Terms of a Finite GP (Sn): The formula Sn = (a1 * (r^n - 1))/(r-1) allows to calculate the sum of a GP up to a certain term. In this formula, a1 represents the first term, r is the GP ratio, and n is the total number of terms.

• Application Examples: Simple and direct demonstrations of how to calculate sums of increasing, decreasing, and constant GPs reinforce the understanding and practical application of the concept.

Conclusions

• Relevance of calculating the sum in a GP: The ability to calculate the sum of the terms of a GP up to a certain point is an essential tool in mathematics and in many areas of everyday life.

• Understanding the relationship between the components of a GP (a1, r, n) and the sum (Sn): The relationship between the first term (a1), the ratio (r), the total number of terms (n), and the sum of the terms (Sn) of a GP is intrinsic and interdependent.

• Exploration of the particularities of increasing, decreasing, and constant GPs: The practical application of calculating the sum of GPs in various scenarios, including increasing, decreasing, and constant GPs, enhances the understanding of the concept and its applicability.

Exercises

1. Calculate the sum of the first 5 terms of the GP (3, 9, 27, 81, ...).

   • Use the formula Sn = (a1 * (r^n - 1))/(r-1). Tip: a1 = 3, r = 3, n = 5.

2. Given the GP (5, 1, 0.2, 0.04, ...), find the sum of the first 4 terms.

   • Remember the formula Sn = (a1 * (r^n - 1))/(r-1). Tip: a1 = 5, r = 0.2, n = 4.

3. Explain and calculate what would be the sum of the first 10 terms of the following GP: (4, 4, 4, 4, ...).

   • Although it is a constant GP, the formula Sn = (a1 * (r^n - 1))/(r-1) does not apply. Find the sum through the relation a1 * n, where a1 is the value of any term in the sequence and n is the number of terms. In this case, a1 = 4, n = 10.