Evaluating The Sum Σ(from I=5 To 12) -i(i-5) A Step-by-Step Guide

In this article, we delve into the evaluation of a specific summation problem. Our focus will be on the summation expression Σ(from i=5 to 12) -i(i-5). This mathematical problem requires us to compute the sum of a series where each term is defined by the expression -i(i-5), and the index i ranges from 5 to 12. This type of problem is commonly encountered in various fields of mathematics, including calculus, discrete mathematics, and numerical analysis. Understanding how to evaluate summations is crucial for solving a wide range of mathematical and computational problems. This article aims to provide a comprehensive, step-by-step approach to solving the given summation, making it accessible to readers with a basic understanding of mathematical notation and operations. Through a clear and detailed explanation, we will break down the complexities of the summation, ensuring that the solution is not only correct but also easily understood. This process will involve expanding the expression, applying summation rules, and performing the necessary calculations. By the end of this article, readers should have a solid understanding of how to evaluate similar summations and appreciate the underlying principles involved.

The summation notation, represented by the Greek letter Σ (sigma), is a compact and powerful way to express the sum of a sequence of terms. The expression Σ(from i=5 to 12) -i(i-5) might seem daunting at first glance, but it becomes clearer when we break it down into its components. The sigma (Σ) indicates that we are performing a summation. The index variable, in this case i, is the variable that changes with each term in the sum. The limits of summation, given as i=5 and 12, specify the range of values that i will take. Specifically, i starts at 5 and increments by 1 until it reaches 12. The expression -i(i-5) is the formula for the terms being added. Each value of i within the specified range is plugged into this formula to generate a term in the sum. To fully grasp this notation, consider a simpler example: Σ(from i=1 to 3) i^2. This means we sum the squares of i for i from 1 to 3, which is 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14. Similarly, in our original problem, we will substitute i with each integer from 5 to 12 into the expression -i(i-5) and add up the results. Understanding the notation is the first crucial step in evaluating the summation correctly. Without a firm grasp of this fundamental concept, attempting to solve the problem would be akin to navigating without a map. The summation notation is not just a shorthand; it is a structured way to represent a series of additions, which is essential in various mathematical contexts.

To evaluate the summation Σ(from i=5 to 12) -i(i-5), we will proceed step by step, ensuring clarity and precision in our calculations. The first step involves expanding the expression inside the summation. The expression -i(i-5) can be expanded to -i^2 + 5i. This algebraic manipulation is crucial as it transforms the expression into a more manageable form for summation. Once the expression is expanded, we can rewrite the original summation as Σ(from i=5 to 12) (-i^2 + 5i). Now, we can apply the properties of summation to break the sum into two separate summations: Σ(from i=5 to 12) (-i^2) + Σ(from i=5 to 12) (5i). This separation makes the summation process easier to handle. Next, we can factor out the constants from each summation, which gives us -Σ(from i=5 to 12) (i^2) + 5Σ(from i=5 to 12) (i). We now have two standard summations: the sum of squares and the sum of integers. We can use known formulas for these summations or compute them directly. For the sum of squares, Σ(from i=5 to 12) (i^2), we need to calculate 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2. Similarly, for the sum of integers, Σ(from i=5 to 12) (i), we calculate 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12. By computing these sums individually and then substituting them back into our expression, we will arrive at the final result. This methodical approach not only ensures accuracy but also helps in understanding the underlying principles of summation. Each step is designed to simplify the problem, making it more approachable and less prone to errors. This step-by-step evaluation is a testament to the power of breaking down complex problems into smaller, more manageable parts.

Following the step-by-step evaluation approach, we now focus on the detailed calculation of the individual sums. The first sum we need to compute is Σ(from i=5 to 12) (i^2), which represents the sum of the squares of the integers from 5 to 12. This can be calculated as follows: 5^2 + 6^2 + 7^2 + 8^2 + 9^2 + 10^2 + 11^2 + 12^2 = 25 + 36 + 49 + 64 + 81 + 100 + 121 + 144. Adding these values together, we get 620. Therefore, Σ(from i=5 to 12) (i^2) = 620. The second sum we need to calculate is Σ(from i=5 to 12) (i), which represents the sum of the integers from 5 to 12. This can be calculated as: 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12. Adding these values together, we get 68. Therefore, Σ(from i=5 to 12) (i) = 68. These calculations are crucial for obtaining the final result of the original summation. Accurate computation of these sums is essential to avoid errors and ensure the correctness of the solution. Now that we have the values for both Σ(from i=5 to 12) (i^2) and Σ(from i=5 to 12) (i), we can substitute these values back into our expression from the previous step. This detailed calculation of sums is a practical application of arithmetic skills and demonstrates the importance of precision in mathematical computations. By breaking down the problem into smaller parts, we can tackle complex summations with confidence and accuracy. The process of calculating these sums also reinforces the understanding of the summation notation and its practical application.

With the detailed calculations of the individual sums completed, we are now ready for the final substitution and determination of the result. Recall that we had broken down the original summation into -Σ(from i=5 to 12) (i^2) + 5Σ(from i=5 to 12) (i). We have already calculated that Σ(from i=5 to 12) (i^2) = 620 and Σ(from i=5 to 12) (i) = 68. Substituting these values into the expression, we get -620 + 5(68). Now, we perform the multiplication: 5 multiplied by 68 equals 340. So, the expression becomes -620 + 340. Finally, we perform the addition: -620 plus 340 equals -280. Therefore, the value of the summation Σ(from i=5 to 12) -i(i-5) is -280. This final substitution and calculation bring together all the previous steps, showcasing the power of methodical problem-solving. The result, -280, is the culmination of expanding the expression, applying summation properties, calculating individual sums, and finally, substituting the values to arrive at the answer. This process not only provides the solution but also reinforces the understanding of summation notation and its applications. The clarity and precision in each step are crucial to avoid errors and ensure the accuracy of the final result. The successful evaluation of this summation demonstrates the importance of a systematic approach in mathematics, where complex problems can be broken down into manageable steps, leading to a clear and concise solution. This final result is not just a number; it is the embodiment of a mathematical journey, from understanding the notation to applying the principles and arriving at the answer.

In conclusion, we have successfully evaluated the summation Σ(from i=5 to 12) -i(i-5) by employing a systematic, step-by-step approach. This journey began with understanding the summation notation, which is the foundation for expressing and solving such problems. We then expanded the expression inside the summation to simplify it, making it more amenable to calculation. By applying the properties of summation, we were able to break the original problem into smaller, more manageable parts, specifically the sum of squares and the sum of integers. This separation allowed us to focus on each component individually, reducing the complexity of the overall problem. We then meticulously calculated the individual sums, ensuring accuracy in our computations. With the values of these sums in hand, we proceeded to the final substitution, where we combined the results to arrive at the final answer. The result of the summation, -280, is a testament to the power of a structured and methodical approach in mathematics. This process not only provides the solution but also enhances our understanding of the underlying principles and techniques involved in summation. By breaking down the problem into smaller, well-defined steps, we were able to tackle a seemingly complex problem with confidence and clarity. The lessons learned from this exercise are applicable to a wide range of mathematical problems, emphasizing the importance of clear notation, strategic simplification, and precise calculation. This article serves as a guide for evaluating similar summations and underscores the value of methodical problem-solving in mathematics. The journey from understanding the notation to arriving at the final result is a clear demonstration of how mathematical concepts can be applied to solve real-world problems and challenges. The summation, often seen as an abstract concept, finds its practical application in various fields, including physics, engineering, and computer science, making its understanding essential for anyone pursuing these disciplines.