Infinite Geometric Series Sum Calculation $a_n=64(1/4)^{n-1}$

Hey everyone! Today, we're diving into the fascinating world of infinite geometric series. Specifically, we're going to tackle the problem of finding the sum (if it exists) of the infinite geometric series related to the infinite geometric sequence described by $a_n=64\left(\frac{1}{4}\right)^{n-1}$. This might sound intimidating, but trust me, it's totally manageable. We'll break it down step-by-step, so you'll be a pro in no time! Get ready to explore the beauty of mathematical sequences and series!

Understanding Geometric Sequences and Series

Before we jump into solving the problem, let's quickly recap what geometric sequences and series are all about. This foundational understanding is absolutely key to grasping the concept of infinite sums. A geometric sequence, at its heart, is a list of numbers where each term is found by multiplying the previous term by a constant value. This constant value, my friends, is what we call the common ratio (r). Think of it like a recipe where you keep scaling the ingredients by the same factor.

For example, the sequence 2, 4, 8, 16... is a geometric sequence because each term is twice the previous term (so, r = 2). Similarly, 100, 50, 25, 12.5... is also a geometric sequence, but here the common ratio is 1/2. Now, a geometric series is simply the sum of the terms in a geometric sequence. So, if our sequence is 2, 4, 8, the corresponding series would be 2 + 4 + 8. Easy peasy, right? The magic really starts to happen when we consider infinite geometric series – series with an infinite number of terms. Can we actually add up infinitely many numbers and get a finite result? That's the million-dollar question we'll be exploring today! The formula for the nth term ($a_n$) of a geometric sequence is given by $a_n = a_1 * r^(n-1)$ where $a_1$ is the first term, r is the common ratio and n is the term number. To fully understand the sum of an infinite geometric series, we'll need to explore the conditions under which such a sum can exist. Intuitively, if the common ratio is too large (in absolute value), the terms might not shrink quickly enough, and the sum could grow without bound, like adding more and more to a pile that just keeps getting bigger. But if the common ratio is small enough, the terms become increasingly tiny, and the sum might approach a finite value. That's the sweet spot we're looking for. This brings us to the concept of convergence. An infinite geometric series converges (i.e., has a finite sum) if and only if the absolute value of the common ratio is less than 1 ( |r| < 1 ). When this condition is met, the sum can be calculated using a specific formula, which we'll get to shortly. Think of this condition as a filter, separating the series that have a finite sum from those that don't. Now that we've laid the groundwork, let's dive into the specific problem at hand. We'll identify the first term and the common ratio of the given sequence, check if the series converges, and then, if it does, we'll calculate its sum. Stay tuned, guys, it's about to get interesting!

Identifying the First Term and Common Ratio

The first step in tackling this problem is to identify the first term ($a_1$) and the common ratio ($r$) of the geometric sequence. Remember, the sequence is defined by the formula $a_n=64\left(\frac{1}{4}\right)^{n-1}$. Let's break it down, guys. The formula itself is a treasure map, leading us directly to the values we need. To find the first term ($a_1$), we simply substitute $n = 1$ into the formula: $a_1=64\left(\frac{1}{4}\right)^{1-1} = 64\left(\frac{1}{4}\right)^0 = 64 * 1 = 64$. So, our first term is a hefty 64. Now, let's hunt down the common ratio ($r$). Looking at the formula, we can see that the term $\left(\frac{1}{4}\right)^{n-1}$ is the key. The base of this exponent, $\frac{1}{4}$, is our common ratio. Think about it: each time 'n' increases by 1, we're multiplying by another factor of $\frac{1}{4}$. That's the very essence of a geometric sequence! So, we have $r = \frac{1}{4}$. Identifying these two key values, $a_1 = 64$ and $r = \frac{1}{4}$, is crucial because they are the building blocks for everything else we'll do. They're like the coordinates on a map, guiding us to the solution. Without them, we'd be wandering in the dark. Once we have the first term and the common ratio, we can start to analyze the behavior of the series. Is it going to converge to a finite sum, or will it zoom off to infinity? This is where the concept of convergence comes into play. Remember, an infinite geometric series converges only if the absolute value of the common ratio is less than 1 ( |r| < 1 ). Let's check if this condition holds true for our series. We have $r = \frac{1}{4}$, and the absolute value of $\frac{1}{4}$ is simply $\frac{1}{4}$, which is definitely less than 1. So, we've got good news! Our series converges, meaning we can actually find a finite sum. This is like reaching the summit of a mountain and realizing the view is even more breathtaking than you imagined. Now that we know the series converges, the next step is to calculate that magical sum. We'll use a specific formula designed just for this purpose. Get ready to see how it all comes together! Before moving on, it's a good idea to pause and reflect on what we've done so far. We've successfully extracted the first term and common ratio from the given formula, and we've confirmed that the series converges. These are significant milestones in our journey. The rest of the problem will build on this foundation, so make sure you're comfortable with these concepts before proceeding. Great job, guys! Let's keep the momentum going!

Calculating the Sum of the Infinite Geometric Series

Now for the grand finale: calculating the sum of our infinite geometric series. This is where all our hard work pays off, and we get to see the beautiful result of adding up infinitely many terms. Remember, we've already established that our series converges, meaning it has a finite sum. This is crucial because not all infinite geometric series have a sum – some of them just keep growing forever! But ours is well-behaved, thanks to its common ratio of $\frac{1}{4}$. So, how do we actually calculate this sum? Well, there's a handy-dandy formula specifically designed for this purpose: $S = \frac{a_1}{1 - r}$. This formula is like a magic key that unlocks the sum of any convergent infinite geometric series. It's elegant, efficient, and incredibly useful. But where does this formula come from? That's a fascinating question that delves into the realm of mathematical proofs and limits. While we won't go into a full-blown derivation here, it's worth knowing that the formula is based on the idea that as you add more and more terms of the series, the sum gets closer and closer to a specific value. That value is what the formula gives us. Now, let's put this formula to work! We already know that our first term, $a_1$, is 64, and our common ratio, $r$, is $\frac{1}{4}$. So, all we need to do is plug these values into the formula: $S = \frac{64}{1 - \frac{1}{4}}$. Let's simplify this expression step-by-step. First, we need to deal with the denominator: $1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}$. So, our formula now looks like this: $S = \frac{64}{\frac{3}{4}}$. To divide by a fraction, we multiply by its reciprocal: $S = 64 * \frac{4}{3} = \frac{64 * 4}{3} = \frac{256}{3}$. Now, let's convert this improper fraction into a mixed number or a decimal. Dividing 256 by 3, we get 85 with a remainder of 1. So, $\frac{256}{3} = 85 \frac{1}{3}$. As a decimal, this is approximately 85.3333... or $85.\overline{3}$. And there you have it! The sum of the infinite geometric series is $85.\overline{3}$. We've successfully navigated the world of infinite sums and emerged victorious. This is a significant achievement, guys. You've not only learned how to calculate the sum of a specific series but also gained a deeper understanding of the underlying concepts of geometric sequences, series, and convergence. But remember, mathematics is not just about getting the right answer. It's about the journey, the process of problem-solving, and the joy of discovery. So, take a moment to appreciate what you've learned today. And don't be afraid to tackle new challenges and explore the endless wonders of mathematics! Now that we have our answer, let's go back and look at the multiple-choice options provided in the original problem. We'll see which one matches our calculated sum and celebrate our final victory!

Matching the Solution to the Options

Alright, let's wrap things up by matching our calculated sum to the multiple-choice options provided. This is the final step in our problem-solving journey, and it's always satisfying to see everything come together. Remember, we found that the sum of the infinite geometric series is $85.\overline{3}$. Now, let's take a look at the options:

A. $S=150 . \overline{2}$ B. $S=85 . \overline{3}$ C. This infinite geometric series does not have a sum.

It's pretty clear, isn't it? Option B, $S=85.\overline{3}$, perfectly matches our calculated sum. That's the correct answer! We did it, guys! We successfully found the sum of the infinite geometric series and identified the correct option. This is a moment to be proud of. You've demonstrated a solid understanding of geometric sequences, series, and the concept of convergence. You've also honed your problem-solving skills, which are valuable in all areas of life. Option A is incorrect, as it's significantly larger than our calculated sum. Option C is also incorrect because we know that the series converges and has a finite sum. The fact that we were able to calculate the sum proves that Option C is false. So, the correct answer is definitively Option B. Before we officially declare victory, let's just recap the entire process we followed. This will help solidify your understanding and make you even more confident in tackling similar problems in the future. We started by understanding the definitions of geometric sequences and series. Then, we identified the first term and common ratio of the given sequence. We checked for convergence and, finding that the series did converge, we applied the formula for the sum of an infinite geometric series. We carefully calculated the sum and finally matched it to the correct option. That's a comprehensive and successful problem-solving approach! Now, you're well-equipped to handle a wide range of problems involving infinite geometric series. Remember, the key is to break down the problem into smaller, manageable steps, understand the underlying concepts, and apply the appropriate formulas and techniques. And most importantly, don't be afraid to ask questions and seek help when you need it. Mathematics is a collaborative journey, and we're all in this together. Congratulations on mastering this problem, guys! Keep up the great work, and I look forward to exploring more mathematical adventures with you in the future!

B. $S=85 . \overline{3}$