Equation For Partial Sum Of Geometric Series Explained

Understanding Geometric Series and Partial Sums

When diving into the world of mathematics, geometric series are a fascinating topic, especially when we explore their partial sums. In essence, a geometric series is a sequence of numbers where each term is multiplied by a constant factor to get the next term. This constant factor is often referred to as the common ratio. Understanding geometric series is crucial in various fields, from finance to physics, as they model scenarios involving exponential growth or decay.

In this article, we will dissect the given geometric series and identify the equation that correctly represents its partial sum. The partial sum of a geometric series is the sum of a finite number of terms in the series. It's a fundamental concept that allows us to analyze the behavior of the series over a specific range. To fully grasp this concept, we need to understand the notation and components involved, such as the first term, the common ratio, and the number of terms we are summing.

The ability to determine the partial sum of a geometric series is not just an academic exercise. It has practical applications in areas like compound interest calculations, where the sum of periodic payments over time forms a geometric series. Similarly, in physics, the decay of radioactive substances can be modeled using geometric series. Therefore, a solid understanding of partial sums enhances our ability to solve real-world problems and make informed decisions.

Before we dive into the specific problem, let's revisit the general formula for the sum of the first n terms of a geometric series. If the first term is denoted by 'a', and the common ratio is 'r', then the sum $S_n$ of the first n terms is given by the formula:

$S_n = a(1 - r^n) / (1 - r)$, where r ≠ 1

This formula is a powerful tool for calculating partial sums, but it's equally important to understand the underlying concept of how these sums are constructed term by term. By exploring the components of the series and their relationships, we can develop a more intuitive understanding of the mathematical principles at play.

Dissecting the Given Geometric Series

The question presents us with the following geometric series expression:

$\\sum_{n=1}^4(125)(\\frac{1}{5})^{n-1}$

This mathematical notation might seem daunting at first, but it's simply a compact way of representing the sum of a series. The symbol $\\sum$ (sigma) indicates summation, and the expression following it defines the terms to be added. Let's break down each component of this expression to make it clearer.

• The index variable: The variable 'n' starts at 1 and goes up to 4. This tells us that we are summing the terms from the first term (n=1) up to the fourth term (n=4). The index variable is the engine that drives the generation of terms in our series.
• The expression inside the summation: $(125)(\\frac{1}{5})^{n-1}$ is the formula that generates each term of the series. This formula tells us exactly how to calculate each term based on the value of 'n'. It's the heart of the geometric series, defining the relationship between successive terms.
• The first term: When n = 1, the term is $(125)(\\frac{1}{5})^{1-1} = (125)(\\frac{1}{5})^0 = 125 * 1 = 125$. The first term is the starting point of our series, and it sets the scale for all subsequent terms.
• The second term: When n = 2, the term is $(125)(\\frac{1}{5})^{2-1} = (125)(\\frac{1}{5})^1 = 125 * \\frac{1}{5} = 25$. Notice how the second term is obtained by multiplying the first term by the common ratio.
• The third term: When n = 3, the term is $(125)(\\frac{1}{5})^{3-1} = (125)(\\frac{1}{5})^2 = 125 * \\frac{1}{25} = 5$. The pattern of multiplying by the common ratio continues, creating the geometric progression.
• The fourth term: When n = 4, the term is $(125)(\\frac{1}{5})^{4-1} = (125)(\\frac{1}{5})^3 = 125 * \\frac{1}{125} = 1$. The fourth term completes the series up to the specified limit, giving us all the components we need to calculate the partial sum.

By carefully evaluating the terms for each value of 'n', we can explicitly write out the series. This process of expansion is crucial for understanding the structure of the series and verifying the partial sum. It's like unpacking a complex machine to see how all the parts fit together.

Identifying the Correct Partial Sum Equation

Now that we've broken down the geometric series, we can determine which equation represents its partial sum. We've calculated the first four terms of the series as 125, 25, 5, and 1. The partial sum, in this case, is simply the sum of these terms.

To find the correct equation, we need to match the terms we've calculated with the options provided. Let's review the options:

• A. 125 + 25 + 5 + 1
• B. 25 + 5 + 1 + $\\frac{1}{5}$
• C. 1 + $\\frac{1}{5}$ + $\\frac{1}{25}$ + $\\frac{1}{125}$
• D. $\\frac{1}{125}$ + $\\frac{1}{5}$ + 5 + 125

Option A, 125 + 25 + 5 + 1, perfectly matches the terms we calculated. This equation represents the sum of the first four terms of the geometric series. The other options either have incorrect terms or do not include the correct sequence of terms.

For instance, Option B includes $\\frac{1}{5}$, which is the fifth term of the series but not part of the sum up to n=4. Option C presents the reciprocals of the terms, and Option D reverses the order and also includes reciprocals. Therefore, only Option A accurately represents the partial sum of the given geometric series.

This exercise highlights the importance of carefully evaluating each term of the series and matching them to the given options. It's a process of verification and elimination, ensuring that we select the equation that precisely reflects the sum we've calculated.

By identifying the correct equation, we demonstrate our understanding of geometric series and their partial sums. This skill is invaluable in more complex mathematical problems and real-world applications.

Conclusion: The Partial Sum Unveiled

In conclusion, the equation that represents the partial sum of the geometric series $\\sum_{n=1}^4(125)(\\frac{1}{5})^{n-1}$ is A. 125 + 25 + 5 + 1. This result is derived by carefully evaluating each term of the series for n = 1 to 4 and then summing those terms.

The process of solving this problem reinforces the fundamental concepts of geometric series and partial sums. We've seen how to break down a summation notation, calculate individual terms, and construct the partial sum. This understanding is crucial for tackling more complex problems involving series and sequences.

Furthermore, this exercise underscores the importance of attention to detail in mathematics. Each term in the series must be calculated accurately, and the final sum must be verified against the given options. A small error in calculation can lead to an incorrect answer, so precision is paramount.

By mastering the concepts of geometric series and partial sums, we equip ourselves with powerful tools for problem-solving in various mathematical and real-world contexts. From calculating compound interest to modeling physical phenomena, the applications of these concepts are vast and varied.

As we continue our mathematical journey, understanding the building blocks of series and sequences will pave the way for exploring more advanced topics. The ability to dissect and analyze series is a cornerstone of mathematical literacy, and it empowers us to tackle a wide range of challenges with confidence and competence.