Calculating Arithmetic Series Sums: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the fascinating world of arithmetic series. Specifically, we'll learn how to find the sum ($S_n$) of an arithmetic series using a handy formula. Let's break down the process, making sure everything is clear and easy to understand. We'll also tackle a specific example to solidify our understanding. So, grab your calculators, and let's get started!

Understanding Arithmetic Series

Arithmetic series are sequences of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference. Think of it like climbing a staircase where each step is the same height. The numbers in the series increase or decrease by a consistent amount each time. Examples of arithmetic series include 2, 4, 6, 8,... (common difference of 2) or 10, 7, 4, 1,... (common difference of -3). Identifying an arithmetic series is crucial because it allows us to use specific formulas to find sums, individual terms, and other properties.

Key Components and Formulas

To find the sum of an arithmetic series, we need a few key components. Firstly, we need to know the number of terms we're summing (n). Secondly, we need the first term ($a_1$) and the last term ($a_n$) in the series. Armed with these, we can utilize the formula: $S_n = rac{n}{2}(a_1 + a_n)$. This formula neatly sums up the series. It's like finding the average of the first and last numbers, then multiplying it by the total count. Another way to view this is by pairing the first and last terms, the second and second-to-last terms, and so on. The sum of each pair is constant, and the formula helps us calculate the total sum efficiently.

Real-World Applications

Arithmetic series isn't just a theoretical concept; they pop up in numerous real-world situations. Think about an investment plan where you deposit a fixed amount each month, or a fitness routine where you increase your running distance by a consistent amount every week. Even in simple tasks, like stacking bricks where each layer has a fixed increment, the principles of arithmetic series apply. They're fundamental to understanding sequences and patterns across various fields, including finance, engineering, and even computer science. Grasping this concept allows us to make predictions, calculate totals, and solve complex problems with ease.

Solving the Problem: A Detailed Walkthrough

Now, let's dive into the core problem: calculating the sum of the given arithmetic series. We're provided with a specific set of parameters, and we will apply our formula to find the correct answer. This section provides a practical, step-by-step approach to ensure you can confidently solve similar problems. We'll use the formula and the information we have to calculate the total sum.

Identifying the Given Values

First, let's identify the given values. We are provided with:

• The first term, $a_1 = 12$
• The last term, $a_n = 75$
• The number of terms, $n = 13$

These values are all we need to get started. Understanding these values is crucial to correctly applying the formula. If any value is missing, the problem cannot be solved using this method, making it essential to have all necessary components. With these in hand, the problem boils down to a plug-and-chug exercise, ensuring we don't need any additional information to arrive at our answer.

Applying the Formula

Now, let's apply the formula: S_n = rac{n}{2}(a_1 + a_n). Substituting the given values, we get: S_{13} = rac{13}{2}(12 + 75). The formula does the heavy lifting, but the arithmetic is our responsibility. Ensure you are substituting each value correctly into its position to avoid calculation errors. Careful substitution is the key to solving the problem with confidence, ensuring we arrive at the right answer.

Calculating the Sum

Let's continue to simplify the expression: S_{13} = rac{13}{2}(87). Now, perform the multiplication: $S_{13} = 6.5 imes 87$. This calculation yields $S_{13} = 565.5$. This step is a straightforward multiplication. You can use a calculator or do it manually, either way ensuring the accuracy of this step is critical. A minor calculation error can lead to a wrong answer, so double-checking your work is important, especially when dealing with such numerical problems.

The Correct Answer and Why

Based on our calculations, the correct answer is $S_{13} = 565.5$. This matches option B in your provided choices. Options A, C, and D are incorrect because they reflect errors in either the formula application or the arithmetic calculation. By carefully following each step, as detailed above, we have arrived at the accurate solution. Understanding how to find each term and its placement within the formula is paramount. Getting the correct answer relies on a sound grasp of the arithmetic series and attention to detail.

Conclusion and Further Learning

In conclusion, we've successfully found the sum of the given arithmetic series. We started by understanding the fundamental properties of the arithmetic series, identified the given values, applied the appropriate formula, and performed the calculations to arrive at the correct answer. The key takeaway here is how to apply the formula effectively. Mastering these concepts provides a solid foundation for more advanced topics in mathematics and real-world applications. Continue practicing similar problems to enhance your understanding and build your confidence. You can also explore other resources such as textbooks, online tutorials, and practice questions to delve deeper into this topic. Math can be fun and challenging, and with consistent effort, you will surely master it! To enhance your knowledge further, consider visiting Khan Academy's Arithmetic Series Section. This website offers additional practice and clear explanations. Keep practicing, and you’ll get better every day!