Evaluate Sum Of Series: Step-by-Step Solution

Evaluating sums, especially those involving series, can seem daunting at first. But fear not, guys! With a solid understanding of the underlying concepts and a step-by-step approach, you can tackle even the most complex summations. In this article, we will dive deep into the evaluation of the sum $\sum_{n=1}^{20} 4\left(\frac{8}{9}\right)^{n-1}$, breaking down each step and ensuring you grasp the core principles involved. We'll explore the world of geometric series, learn how to identify them, and apply the appropriate formulas to arrive at the solution. So, buckle up and let's embark on this mathematical journey together!

Understanding Geometric Series

Before we jump into the specifics of the problem, let's take a moment to refresh our understanding of geometric series. A geometric series is a sequence of numbers where each term is multiplied by a constant value called the common ratio to get the next term. Think of it like this: you start with a number, and then you repeatedly multiply it by the same factor. This simple pattern gives rise to some pretty cool mathematical properties. These series pop up in various fields, from finance (think compound interest) to physics (like radioactive decay), making them an essential concept to master. The general form of a geometric series is: a, ar, ar^2, ar^3, ..., where 'a' is the first term and 'r' is the common ratio. The sum of the first 'n' terms of a geometric series can be calculated using a nifty formula, which we'll get to in a bit. Recognizing a geometric series is the first step to solving many problems, so keep an eye out for that constant multiplication pattern. Identifying the first term and common ratio are the next critical steps to utilize the formula. It is very important to know these values as they directly affect the solution when dealing with geometric series, impacting the final sum. Don't worry, we'll walk through examples to make sure you've got it down pat.

Identifying the Geometric Series in the Problem

Now that we've got a handle on geometric series in general, let's zoom in on the specific sum we need to evaluate: $\sum_{n=1}^{20} 4\left(\frac{8}{9}\right)^{n-1}$. The first step is to confirm that this is indeed a geometric series. Looking at the expression, we see that the term $4\left(\frac{8}{9}\right)^{n-1}$ has a constant base (8/9) raised to a power that depends on 'n'. This is a telltale sign of a geometric series. As 'n' increases, each term is multiplied by the same factor, which is our common ratio. To be absolutely sure, we can write out the first few terms of the series by plugging in n = 1, 2, 3, and so on. When n = 1, the term is 4*(8/9)^(1-1) = 4*(8/9)^0 = 41 = 4. When n = 2, the term is 4(8/9)^(2-1) = 4*(8/9)^1 = 4*(8/9) = 32/9. When n = 3, the term is 4*(8/9)^(3-1) = 4*(8/9)^2 = 4*(64/81) = 256/81. We can see that each term is obtained by multiplying the previous term by 8/9, confirming our suspicion. The first term (a) of this series is 4, which we found when n = 1. The common ratio (r) is 8/9, the constant factor by which each term is multiplied. Now that we've identified 'a' and 'r', we're well on our way to calculating the sum!

Applying the Formula for the Sum of a Geometric Series

Alright, we've successfully identified our series as geometric, and we've pinpointed the first term (a = 4) and the common ratio (r = 8/9). Now comes the fun part: applying the formula to calculate the sum of the first 'n' terms of a geometric series. The formula is a powerful tool that allows us to quickly find the sum without having to manually add up each term. Here's the formula: S_n = a(1 - r^n) / (1 - r), where S_n is the sum of the first 'n' terms, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms. This formula might look a bit intimidating at first, but it's actually quite straightforward to use. Just plug in the values you know, and the rest is just arithmetic! In our case, we want to find the sum of the first 20 terms (n = 20) of the series $\sum_{n=1}^{20} 4\left(\frac{8}{9}\right)^{n-1}$. We already know that a = 4 and r = 8/9. So, we can substitute these values into the formula: S_{20} = 4(1 - (8/9)^{20}) / (1 - 8/9). Now it's just a matter of simplifying this expression. Don't worry, we'll take it step by step.

Calculating the Sum Step-by-Step

Let's break down the calculation of the sum S_20} = 4(1 - (8/9)^{20}) / (1 - 8/9) step by step. This will help us avoid any calculation errors and ensure we arrive at the correct answer. First, let's tackle the denominator (1 - 8/9). This is a simple subtraction: 1 - 8/9 = 9/9 - 8/9 = 1/9. So, our equation now looks like this: S_{20 = 4(1 - (8/9)^20}) / (1/9). Next, let's focus on the term (8/9)^{20}. This is where a calculator comes in handy. Calculating (8/9)^{20} gives us approximately 0.09476. So, we have S_{20 = 4(1 - 0.09476) / (1/9). Now, let's simplify the expression inside the parentheses: 1 - 0.09476 = 0.90524. Our equation is now: S_20} = 4 * 0.90524 / (1/9). Next, let's multiply 4 by 0.90524 4 * 0.90524 = 3.62096. So, we have: S_{20 = 3.62096 / (1/9). Finally, dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1/9 is 9. So, S_20} = 3.62096 * 9. Multiplying these values gives us S_{20 ≈ 32.58864. Therefore, the sum of the first 20 terms of the series is approximately 32.58864. We've successfully evaluated the sum! It's always a good idea to double-check your work, especially when dealing with multiple steps. Make sure you've correctly substituted the values into the formula and that you've performed the arithmetic operations accurately. A small error in one step can lead to a big difference in the final answer.

Conclusion: Mastering Geometric Series

Woohoo! We've successfully navigated the world of geometric series and evaluated the sum $\sum_{n=1}^{20} 4\left(\frac{8}{9}\right)^{n-1}$. We've seen how to identify a geometric series, determine its first term and common ratio, and apply the formula for the sum of its first 'n' terms. This is a powerful set of skills that will serve you well in your mathematical journey. Remember, the key to mastering these concepts is practice. Work through different examples, and don't be afraid to make mistakes. Each mistake is a learning opportunity. With a little bit of effort, you'll be able to tackle any geometric series that comes your way. So, keep practicing, keep exploring, and keep having fun with math! This stuff is super useful in various real-world applications, from calculating investments to modeling physical phenomena. The more you understand these concepts, the better equipped you'll be to solve problems and make informed decisions. So, congratulations on mastering this topic, and keep up the awesome work!