Evaluating The Series Sum $ \sum_{n=1}^5 3(-2)^{n-1} $ A Comprehensive Guide
Evaluating series can sometimes seem daunting, especially when dealing with exponents and alternating signs. In this comprehensive guide, we will tackle the problem of evaluating the sum $ \sum_{n=1}^5 3(-2)^{n-1} $ step by step, providing clear explanations and insights along the way. This problem involves calculating the sum of a geometric series, a fundamental concept in mathematics with applications ranging from finance to physics. Our goal is not only to find the numerical answer but also to understand the underlying principles and techniques involved in solving such problems. By breaking down the summation and carefully analyzing each term, we can arrive at the solution and gain a deeper appreciation for the elegance of geometric series.
Understanding the Summation Notation
Before we dive into the calculations, let's first understand the summation notation. The expression $ \sum_{n=1}^5 3(-2)^{n-1} $ represents the sum of a sequence of terms. The symbol $ \sum $ (sigma) indicates summation, and the variable $ n $ is the index of summation. The limits of summation are specified below and above the sigma symbol. In this case, $ n $ starts at 1 and ends at 5. This means we need to evaluate the expression $ 3(-2)^{n-1} $ for each integer value of $ n $ from 1 to 5 and then add up the results. Understanding this notation is crucial as it provides a concise way to represent sums of sequences, which are common in various mathematical and scientific contexts. By grasping the notation, we can systematically approach the problem, ensuring that we consider each term in the series correctly and avoid any errors in the calculation. The summation notation is a powerful tool in mathematics, allowing us to express complex sums in a compact and manageable form, which is essential for both theoretical analysis and practical applications.
Expanding the Series
To evaluate the sum, we need to expand the series by substituting each value of $ n $ from 1 to 5 into the expression $ 3(-2)^{n-1} $. This means we will calculate five terms and then add them together. Let's start with $ n = 1 $:
- For $ n = 1 $, the term is $ 3(-2)^{1-1} = 3(-2)^0 = 3(1) = 3 $.
Next, we calculate the term for $ n = 2 $:
- For $ n = 2 $, the term is $ 3(-2)^{2-1} = 3(-2)^1 = 3(-2) = -6 $.
Continuing this process for $ n = 3 $:
- For $ n = 3 $, the term is $ 3(-2)^{3-1} = 3(-2)^2 = 3(4) = 12 $.
Now, let's find the term for $ n = 4 $:
- For $ n = 4 $, the term is $ 3(-2)^{4-1} = 3(-2)^3 = 3(-8) = -24 $.
Finally, we calculate the term for $ n = 5 $:
- For $ n = 5 $, the term is $ 3(-2)^{5-1} = 3(-2)^4 = 3(16) = 48 $.
By systematically substituting each value of $ n $, we have expanded the series into its individual terms. This step is crucial because it allows us to visualize the series and perform the addition more easily. The alternating signs and increasing magnitudes of the terms indicate that this is a geometric series with a negative common ratio. Expanding the series is a fundamental technique in evaluating summations, as it transforms the abstract notation into a concrete sequence of numbers that can be directly manipulated. This process ensures that we account for each term in the series correctly, paving the way for accurate calculation of the sum. The expanded series now looks like this: $ 3 - 6 + 12 - 24 + 48 $.
Identifying the Geometric Series
The expanded series $ 3 - 6 + 12 - 24 + 48 $ is a geometric series. A geometric series is a sequence of numbers where each term is multiplied by a constant value to get the next term. This constant value is called the common ratio. In our series, we can see that each term is obtained by multiplying the previous term by -2. To confirm this, we can divide any term by its preceding term:
- $ -6 / 3 = -2 $
- $ 12 / -6 = -2 $
- $ -24 / 12 = -2 $
- $ 48 / -24 = -2 $
Since the ratio between consecutive terms is constant (-2), we can confidently identify this as a geometric series. Recognizing the series as geometric is essential because it allows us to use specific formulas and techniques to calculate the sum more efficiently. The first term of the series, denoted by $ a $, is 3, and the common ratio, denoted by $ r $, is -2. These values are crucial for applying the formula for the sum of a finite geometric series. Identifying the geometric nature of the series not only simplifies the calculation but also provides insights into the behavior of the sequence. Geometric series are fundamental in mathematics and have numerous applications in various fields, making their recognition and understanding vital for problem-solving. The ability to identify a geometric series allows us to leverage powerful tools and formulas, turning a potentially complex summation into a straightforward calculation.
Calculating the Sum
Now that we have expanded the series and identified it as a geometric series, we can calculate the sum. The sum $ S_n $ of the first $ n $ terms of a geometric series is given by the formula:
$ S_n = a rac{1 - r^n}{1 - r} $,
where $ a $ is the first term, $ r $ is the common ratio, and $ n $ is the number of terms. In our case, $ a = 3 $, $ r = -2 $, and $ n = 5 $. Plugging these values into the formula, we get:
$ S_5 = 3 rac{1 - (-2)^5}{1 - (-2)} $.
First, we need to calculate $ (-2)^5 $:
$ (-2)^5 = -32 $.
Now, substitute this value back into the formula:
$ S_5 = 3 rac{1 - (-32)}{1 - (-2)} = 3 rac{1 + 32}{1 + 2} = 3 rac{33}{3} $.
Simplify the expression:
$ S_5 = 3 rac{33}{3} = 3(11) = 33 $.
Therefore, the sum of the series is 33. This calculation demonstrates the power of the formula for the sum of a geometric series, allowing us to find the sum of a sequence of terms quickly and accurately. By correctly identifying the parameters of the series ($ a $, $ r $, and $ n $) and applying the formula, we can avoid the tedious process of manually adding each term. The formula encapsulates the underlying mathematical structure of the series, providing a concise and efficient method for calculating the sum. The result, 33, is the final answer to our problem, and the process of arriving at this solution highlights the importance of understanding geometric series and their properties. This calculation is a testament to the elegance and efficiency of mathematical formulas in solving complex problems.
Verifying the Result
To verify our result, we can manually add the terms of the series we expanded earlier: $ 3 - 6 + 12 - 24 + 48 $. Let's perform the addition step by step:
- $ 3 - 6 = -3 $
- $ -3 + 12 = 9 $
- $ 9 - 24 = -15 $
- $ -15 + 48 = 33 $
As we can see, the sum of the series is indeed 33, which matches the result we obtained using the formula for the sum of a geometric series. This verification step is crucial because it ensures that our calculations are correct and that we have not made any errors in applying the formula or in the arithmetic. Manual verification provides a sense of confidence in our answer and reinforces our understanding of the problem-solving process. It also highlights the consistency between the expanded form of the series and the concise formula, demonstrating the power and accuracy of mathematical tools. Verifying the result is a best practice in mathematics, as it helps to catch any mistakes and ensures that the final answer is reliable. The agreement between the manual calculation and the formulaic result solidifies our understanding of the geometric series and its summation, confirming that our approach and solution are correct.
Conclusion
In conclusion, we have successfully evaluated the sum $ \sum_{n=1}^5 3(-2)^{n-1} $ by first expanding the series, identifying it as a geometric series, applying the formula for the sum of a finite geometric series, and verifying the result. The sum of the series is 33. This problem demonstrates the importance of understanding summation notation, geometric series, and the formulas associated with them. By breaking down the problem into smaller steps and carefully applying the appropriate techniques, we can solve complex mathematical problems with confidence. The process of evaluating this series has provided valuable insights into the nature of geometric series and their applications. Understanding these concepts is crucial for further studies in mathematics and related fields. The ability to recognize and manipulate series is a fundamental skill that is widely applicable in various areas of science and engineering. This step-by-step guide has not only provided a solution to the specific problem but has also reinforced the underlying principles and techniques involved in evaluating summations, equipping us with the tools to tackle similar problems in the future. The successful evaluation and verification of the sum highlight the power and elegance of mathematical methods in solving complex problems, emphasizing the importance of a systematic and thorough approach.
