Calculating The Sum Of A Geometric Series

Hey everyone! Let's dive into a cool math problem today. We're going to figure out the sum of a series, specifically $\sum_{k=1}^5 2^k$. This might look a little intimidating at first glance, but trust me, it's totally manageable. We'll break it down step by step, so even if you're not a math whiz, you'll get the hang of it. So, grab your coffee (or your favorite beverage), and let's get started. We will explore the concept of geometric series, understand the notation, and ultimately, calculate the sum. This kind of problem often pops up in various fields, from computer science to finance, so understanding it is super valuable.

Understanding the Problem: Unpacking the Summation Notation

First things first, let's decode what $\sum_{k=1}^5 2^k$ actually means. The symbol $\sum$ is the summation symbol. It tells us we need to add a bunch of terms together. The 'k' is our index, and it starts at 1 (the number below the summation symbol) and goes up to 5 (the number above the summation symbol). For each value of 'k' from 1 to 5, we're going to calculate $2^k$ and add it to our running total. So, essentially, we need to calculate $2^1 + 2^2 + 2^3 + 2^4 + 2^5$. See? Not so scary once you break it down! This notation is super common in math and is a concise way to represent adding up a series of terms. It's like a shorthand for a longer expression. Being comfortable with this notation is key to tackling more complex problems. It's all about recognizing the pattern and understanding what each part of the notation signifies. Remember, the lower number tells you where to start, the upper number tells you where to stop, and the expression next to the summation symbol tells you what to calculate for each step. This foundational understanding sets the stage for more complex mathematical concepts.

Let's break down the individual terms. When k = 1, we have $2^1 = 2$. When k = 2, we have $2^2 = 4$. For k = 3, we get $2^3 = 8$. When k = 4, $2^4 = 16$, and finally, when k = 5, $2^5 = 32$. Now, all we need to do is add these values together: 2 + 4 + 8 + 16 + 32.

Calculating the Sum: Step-by-Step Approach

Now that we know what the summation notation means and have expanded the series, the next step is to calculate the sum. We're dealing with a geometric series here, which means each term is multiplied by a constant factor to get the next term. In this case, each term is multiplied by 2 (the base of the exponent). There are a couple of ways to approach this. We can simply add the terms together, or we can use the formula for the sum of a geometric series. Let's do it the straightforward way first. We have 2 + 4 + 8 + 16 + 32. Adding these numbers gives us 62. So, the sum of the series is 62. See? Not too bad, right? We can do this either by simple addition or by using a formula. We will verify the final solution with the formula. This is the simplest way to find the sum of this particular series.

For simple series like this, directly adding the terms is perfectly fine, and often the most efficient method. However, as the series grows longer, or if we need to calculate the sum of many such series, having a general formula comes in handy. It's like having a shortcut. It saves time and minimizes the chance of making calculation errors. Therefore, let's explore the formula as well, so you'll be familiar with it.

The Formula for the Sum of a Geometric Series

For a geometric series, the formula to calculate the sum is: $S_n = a * (1 - r^n) / (1 - r)$, 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. In our case, 'a' is 2 (the first term, $2^1$), 'r' is 2 (the common ratio, since each term is multiplied by 2), and 'n' is 5 (the number of terms). Plugging these values into the formula, we get: $S_5 = 2 * (1 - 2^5) / (1 - 2)$. Let's calculate this: $2^5$ is 32, so the equation becomes $S_5 = 2 * (1 - 32) / (1 - 2)$, simplifying further to $S_5 = 2 * (-31) / (-1)$, which equals $S_5 = -62 / -1$, and finally, $S_5 = 62$. Voila! We get the same answer: 62. Knowing this formula is super useful, especially for longer series. It can save a ton of time and effort. It is a good idea to know how to use the formula and when to use it.

The formula for the sum of a geometric series is a powerful tool. It allows us to calculate the sum without manually adding each term. This is especially useful when dealing with a large number of terms. The formula not only saves time but also reduces the chance of making arithmetic errors. When you understand the formula, you can apply it in many scenarios.

Conclusion and Summary of Concepts

So, to recap, the sum of the series $\sum_{k=1}^5 2^k$ is 62. We found this by expanding the summation, identifying the series as geometric, adding up the terms directly, and also using the formula for the sum of a geometric series. We've gone from a seemingly complex notation to a simple, concrete answer. The key takeaways here are understanding the summation notation, recognizing geometric series, and knowing how to apply the formula if needed. These skills are fundamental in mathematics and have applications in various fields like finance, physics, and computer science. Keep practicing, and you will become super comfortable with these concepts. You've now conquered a problem that probably seemed a bit intimidating at first glance. Awesome work, everyone!

This method is super useful for other mathematical operations. Remember, practice makes perfect. The more you work with these concepts, the more comfortable and confident you'll become. Each problem you solve builds your understanding and your ability to tackle more advanced mathematical challenges. You've now added another tool to your math toolkit, and you're ready to take on even more interesting problems.

The Correct Answer

The correct answer is B. 62.