Calculating The Sum Of A Geometric Series A Step-by-Step Solution

This article delves into the fascinating world of geometric series, providing a comprehensive guide to understanding and calculating their sums. We will specifically address the problem of finding the sum of the geometric series $\sum_{n=1}^4(-2)(-3)^{n-1}$. This problem serves as a great example to illustrate the concepts and techniques involved in summing geometric series. Whether you're a student grappling with series in mathematics or simply someone looking to expand your knowledge, this guide will equip you with the tools and understanding you need. Let's embark on this journey together and unravel the mysteries of geometric series!

Decoding Geometric Series

In the realm of mathematics, geometric series are a fundamental concept, and the sum of a geometric series holds significant importance. Before we dive into solving the specific problem, let's first define what a geometric series is and explore its key components. A geometric series is a series where each term is multiplied by a constant factor to obtain the next term. This constant factor is known as the common ratio, often denoted by 'r'. The general form of a geometric series can be expressed as:

a + ar + ar^2 + ar^3 + ...

Where:

• 'a' represents the first term of the series.
• 'r' is the common ratio.
• The ellipsis (...) indicates that the series can continue infinitely.

Understanding the first term ('a') and the common ratio ('r') is crucial for analyzing and calculating the sum of a geometric series. The common ratio dictates the behavior of the series – whether it converges (approaches a finite sum) or diverges (grows infinitely). When |r| < 1, the geometric series converges, and we can calculate its sum using a specific formula. Conversely, when |r| ≥ 1, the series diverges, and it does not have a finite sum. In the context of a finite geometric series, like the one presented in the problem, we are dealing with a specific number of terms, and the sum can always be calculated regardless of the value of 'r'. Identifying 'a' and 'r' is the first step toward unraveling the sum of any geometric series, and we will apply this understanding to solve our example problem.

Finite Geometric Series

Focusing on finite geometric series is essential, as these are the type of series we encounter in practical problem-solving. A finite geometric series is a series with a limited number of terms. Unlike infinite geometric series, which can extend indefinitely, finite series have a defined starting point and ending point. This makes calculating their sums more straightforward. The general form of a finite geometric series with 'n' terms is:

a + ar + ar^2 + ar^3 + ... + ar^(n-1)

Where:

• 'a' is the first term.
• 'r' is the common ratio.
• 'n' is the number of terms in the series.

The sum of a finite geometric series, often denoted as S_n, can be calculated using a specific formula, which we will explore in detail later. This formula is a powerful tool that allows us to efficiently determine the sum without having to manually add each term. Recognizing the series as finite is the first step, followed by identifying the values of 'a', 'r', and 'n'. These values are then plugged into the sum formula to obtain the final result. In the context of the given problem, the series $\sum_{n=1}^4(-2)(-3)^{n-1}$ is a finite geometric series, as it has a defined number of terms (4 terms, specifically). This characteristic makes it amenable to the sum formula, and we can confidently calculate its sum using the appropriate techniques.

Formula for the Sum

The formula for the sum of a finite geometric series is a cornerstone in dealing with these mathematical constructs. This formula provides a direct and efficient way to calculate the sum without the need for tedious term-by-term addition. The formula is given by:

S_n = a(1 - r^n) / (1 - r)

Where:

• S_n represents the sum of the first 'n' terms of the series.
• 'a' is the first term.
• 'r' is the common ratio.
• 'n' is the number of terms.

This formula is derived using algebraic manipulation and is applicable as long as the common ratio 'r' is not equal to 1. When r = 1, the series becomes a simple arithmetic series, and a different formula is used to calculate the sum. The power of this formula lies in its ability to encapsulate the entire series into a single expression, allowing for quick calculations. To use the formula effectively, it is crucial to correctly identify the values of 'a', 'r', and 'n' from the given series. Once these values are known, they can be substituted into the formula, and the sum S_n can be readily computed. The formula is a valuable tool in various applications, from financial calculations to physics problems, making it an indispensable part of mathematical knowledge.

Applying the Formula to the Problem

Now, let's tackle the specific problem at hand: finding the sum of the geometric series $\sum_{n=1}^4(-2)(-3)^{n-1}$. This requires us to apply the formula we discussed earlier. The first step is to identify the values of 'a', 'r', and 'n' from the given series. By carefully examining the series, we can determine:

• The first term, 'a': When n = 1, the term is (-2)(-3)^(1-1) = (-2)(-3)^0 = (-2)(1) = -2. So, a = -2.
• The common ratio, 'r': The base of the exponent is -3, which is the common ratio. So, r = -3.
• The number of terms, 'n': The series sums from n = 1 to n = 4, indicating there are 4 terms. So, n = 4.

With these values identified, we can now substitute them into the formula for the sum of a finite geometric series:

S_n = a(1 - r^n) / (1 - r)

Substituting a = -2, r = -3, and n = 4, we get:

S_4 = (-2)(1 - (-3)^4) / (1 - (-3))

This substitution sets the stage for the final calculation, where we will evaluate the expression to determine the sum of the series. By breaking down the problem into these steps, we make the process more manageable and reduce the chances of errors.

Calculation Steps

With the values substituted into the sum formula, we proceed with the calculation in a step-by-step manner to ensure accuracy. We have:

S_4 = (-2)(1 - (-3)^4) / (1 - (-3))

First, we evaluate (-3)^4:

(-3)^4 = (-3) * (-3) * (-3) * (-3) = 81

Now, substitute this back into the equation:

S_4 = (-2)(1 - 81) / (1 - (-3))

Next, simplify the terms inside the parentheses:

S_4 = (-2)(-80) / (1 + 3)

S_4 = (-2)(-80) / 4

Now, perform the multiplication in the numerator:

S_4 = 160 / 4

Finally, divide to get the sum:

S_4 = 40

Therefore, the sum of the geometric series $\sum_{n=1}^4(-2)(-3)^{n-1}$ is 40. This result aligns with one of the given options, confirming our solution. The step-by-step calculation illustrates the importance of order of operations and careful arithmetic in arriving at the correct answer. By breaking down the problem into smaller, manageable steps, we can confidently navigate through the calculations and arrive at the solution.

Conclusion

In conclusion, we have successfully determined the sum of the given geometric series, $\sum_{n=1}^4(-2)(-3)^{n-1}$, to be 40. This was achieved by first understanding the fundamental concept of a geometric series, identifying the key components such as the first term (a), the common ratio (r), and the number of terms (n). We then applied the formula for the sum of a finite geometric series, S_n = a(1 - r^n) / (1 - r), after carefully substituting the identified values. The step-by-step calculation process demonstrated how to methodically evaluate the expression and arrive at the correct result. This exercise not only provides the answer to the specific problem but also reinforces the understanding of geometric series and their summation. The ability to work with geometric series is a valuable skill in mathematics and has applications in various fields, including finance, physics, and computer science. By mastering the concepts and techniques presented in this guide, you are well-equipped to tackle similar problems and further explore the fascinating world of mathematical series.