Sum Of Arithmetic Series: Step-by-Step Calculation

Hey guys! Today, we're diving into the fascinating world of arithmetic series. Specifically, we're going to tackle a common problem: finding the sum of an arithmetic series. Don't worry, it's not as scary as it sounds! We'll break it down step by step, so you'll be a pro in no time. We will use the given parameters $a_1 = -68$, $d = -3$, and $n = 20$ to illustrate the process. Let's get started!

Understanding Arithmetic Series

Before we jump into calculations, let's make sure we're all on the same page. An arithmetic series is simply the sum of the terms in an arithmetic sequence. An arithmetic sequence, in turn, is a sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference, often denoted by 'd'. In our case, we have $d = -3$, which means each term is 3 less than the term before it. Think of it like counting down by 3s, but starting from -68! Understanding this foundational concept is crucial because it provides the context for why the formulas we use work. We aren't just plugging numbers into a magical equation; we're applying a logical process that stems from the very nature of arithmetic sequences. For example, if we were to write out the first few terms of this sequence, we'd see something like: -68, -71, -74, -77, and so on. Each term is indeed 3 less than the previous one. This constant difference is the key to unlocking the sum of the series, as it allows us to predict the pattern and ultimately calculate the total. So, keep that constant difference in mind as we move forward, as it's the backbone of our calculations.

Identifying the Key Components

Now that we have a good grasp of what an arithmetic series is, let's pinpoint the key information we need to solve our problem. We're given three crucial pieces of data: the first term ($a_1$), the common difference ($d$), and the number of terms ($n$). These are the building blocks we'll use to construct our solution. The first term, $a_1$, is the starting point of our sequence. In this case, $a_1 = -68$. Think of it as the anchor of our series. The common difference, $d$, as we discussed earlier, is the constant amount by which each term changes. Here, $d = -3$, indicating a decreasing sequence. The number of terms, $n$, tells us how many numbers we're adding together in our series. We have $n = 20$, meaning we're summing the first 20 terms of this arithmetic sequence. It's essential to correctly identify these components because using the wrong value for any of them will lead to an incorrect sum. Imagine mixing up the first term with the common difference – you'd be starting at the wrong point and changing by the wrong amount, throwing off the entire calculation! So, always double-check that you've correctly identified $a_1$, $d$, and $n$ before proceeding. These values are the foundation upon which we'll build our solution, and accuracy here is paramount.

The Formula for the Sum

The good news is, there's a handy formula that lets us calculate the sum of an arithmetic series directly, without having to add up all 20 terms individually. This formula is a lifesaver, especially when dealing with a large number of terms! The formula is: $S_n = \frac{n}{2}[2a_1 + (n-1)d]$, where $S_n$ represents the sum of the first 'n' terms. This formula might look a bit intimidating at first glance, but don't worry, it's actually quite straightforward once you understand what each part represents. Let's break it down piece by piece. $S_n$ is what we're trying to find – the sum of the first 'n' terms. 'n' is the number of terms, which we already know is 20 in our case. $a_1$ is the first term, which is -68. And 'd' is the common difference, which is -3. See? We already have all the pieces of the puzzle! The beauty of this formula lies in its efficiency. Instead of manually adding 20 numbers together, we can simply plug in our values and perform a few calculations. This is especially helpful when dealing with series that have hundreds or even thousands of terms. So, let's embrace this formula and see how it simplifies our task of finding the sum of our arithmetic series.

Plugging in the Values

Now comes the fun part – substituting our values into the formula! We have $a_1 = -68$, $d = -3$, and $n = 20$. Let's plug these into the formula: $S_{20} = \frac{20}{2}[2(-68) + (20-1)(-3)]$. See how we've simply replaced the variables with their corresponding values? This is a crucial step in solving any mathematical problem – ensuring that you've correctly substituted the given information. A common mistake is to accidentally swap values or misread a number, so take your time and double-check your work. Now, let's simplify the expression step by step. First, we have $\frac{20}{2}$, which simplifies to 10. Next, we have $2(-68)$, which equals -136. Then, we have $(20-1)$, which is 19, and we multiply that by -3 to get -57. So, our equation now looks like this: $S_{20} = 10[-136 + (-57)]$. We're making progress! By carefully plugging in the values and breaking down the expression into smaller, manageable parts, we're setting ourselves up for a successful calculation. Remember, accuracy is key, so always double-check each step as you go. We're almost there – just a few more calculations to get to our final answer!

Step-by-Step Calculation

Alright, let's continue simplifying our equation. We have $S_{20} = 10[-136 + (-57)]$. First, let's tackle the expression inside the brackets. We need to add -136 and -57. Remember, adding two negative numbers is like moving further into the negative side of the number line. So, -136 plus -57 equals -193. Now our equation looks like this: $S_{20} = 10[-193]$. We're almost home! The final step is to multiply 10 by -193. Multiplying by 10 is easy – we just add a zero to the end of the number. Since we're multiplying a positive number by a negative number, the result will be negative. So, 10 times -193 equals -1930. Therefore, $S_{20} = -1930$. We've done it! By carefully following each step and performing the calculations accurately, we've successfully found the sum of the first 20 terms of our arithmetic series. It's important to take your time and break down the problem into smaller, manageable steps. This not only helps prevent errors but also makes the process less daunting. So, celebrate your accomplishment – you've mastered the art of calculating the sum of an arithmetic series!

The Final Answer

So, drumroll please... the sum of the arithmetic series with $a_1 = -68$, $d = -3$, and $n = 20$ is -1930. We've successfully navigated through the formula, plugged in the values, and performed the calculations to arrive at our final answer. This result tells us that if we were to add up the first 20 terms of this sequence (-68, -71, -74, and so on), we would get a total of -1930. Isn't that neat? It's like uncovering a hidden pattern within the numbers. But more than just arriving at the answer, it's important to understand the process we went through. We didn't just magically pull -1930 out of thin air. We started with a clear understanding of arithmetic series, identified the key components, applied the correct formula, and carefully executed the calculations. This systematic approach is what truly empowers us to solve similar problems in the future. So, the next time you encounter an arithmetic series problem, remember the steps we took here. You've got the tools and the knowledge to conquer it! And who knows, maybe you'll even start seeing arithmetic series everywhere you go – in patterns, in nature, even in everyday situations. Math is all around us, and now you're one step closer to unlocking its secrets.

Real-World Applications

You might be thinking,