Infinite Geometric Series Sum: 8 + 16/3 + 32/9 + ...

by Dimemap Team 53 views

Hey guys! Let's dive into the fascinating world of infinite geometric series and tackle this problem step-by-step. We're given the series 8 + 16/3 + 32/9 + ... and our mission, should we choose to accept it, is to find the sum of this never-ending sequence. Don't worry, it's not as daunting as it sounds! Understanding geometric series is super important in math, finding applications in physics, engineering, and even economics. So let's get started and break this down.

Understanding Geometric Series

First things first, what exactly is a geometric series? A geometric series is a sequence of numbers where each term is multiplied by a constant value to get the next term. This constant value is called the common ratio, often denoted as 'r'. In our given series, 8 + 16/3 + 32/9 + ..., we need to figure out this common ratio. To identify this common ratio (r), divide any term by its preceding term.

  • r = (16/3) / 8 = 2/3
  • r = (32/9) / (16/3) = 2/3

So, our common ratio, r, is 2/3. This means each term is multiplied by 2/3 to get the next term. Knowing the common ratio is the key to unlocking the sum of this infinite series. Remember, the formula we're about to use works only when the absolute value of r is less than 1 (|r| < 1). Otherwise, the series diverges, meaning it doesn't converge to a finite sum. Thankfully, in our case, |2/3| < 1, so we're good to go!

The Formula for the Sum of an Infinite Geometric Series

Now for the magic formula! The sum (S) of an infinite geometric series is given by:

S = a / (1 - r)

Where:

  • S is the sum of the infinite series.
  • a is the first term of the series.
  • r is the common ratio.

This formula is a cornerstone in understanding how infinite sums can actually have finite values. It elegantly captures the concept that as the series progresses, the terms become increasingly smaller, contributing less and less to the overall sum. This is contingent on the common ratio being a fraction, as that ensures the terms diminish over time. The formula provides a direct and efficient way to calculate the sum without having to add up an infinite number of terms, which is, of course, impossible to do manually. It's a neat trick that mathematics offers us, turning the infinite into something we can grasp and calculate.

Applying the Formula to Our Series

Let's plug in the values from our series into the formula:

  • a (the first term) = 8
  • r (the common ratio) = 2/3

So, the sum S is:

S = 8 / (1 - 2/3)

Let's simplify this:

S = 8 / (1/3)

S = 8 * 3

S = 24

Oops! It seems we made a calculation error somewhere. Let's go back and double-check our work. It’s crucial to always verify each step, especially when dealing with fractions and division. A small mistake can lead to a completely different answer. In mathematics, precision is key, and taking the time to re-evaluate ensures accuracy. Let's meticulously review our steps to pinpoint where we might have slipped up. Let's recalculate:

S = 8 / (1 - 2/3)

S = 8 / (1/3)

S = 8 * 3

S = 24

Okay, we still arrive at 24. Looking at the options provided (A. 48, B. 3, C. 4, D. 5, E. 6), 24 isn't one of them. This suggests there might be an error in the question itself or the provided options. It's not uncommon for discrepancies to occur in problem sets, so it's important to recognize when our calculations don't align with the given answers. In such cases, it's good practice to re-examine the problem statement and our calculations, but also to acknowledge the possibility of an error in the source material. Sometimes, the most valuable skill in problem-solving isn't just arriving at an answer, but also critically evaluating the context and identifying potential inconsistencies.

Double Checking the Options and Identifying the Discrepancy

Let's analyze the options provided again: A. 48, B. 3, C. 4, D. 5, E. 6. None of these match our calculated sum of 24. This strengthens our suspicion that there might be an error in the question or the options. When faced with a situation like this, it's important to trust our calculations and reasoning, especially after verifying each step. In mathematical problem-solving, it's not just about getting the right answer, but also about the process and understanding the underlying concepts. We've demonstrated a clear understanding of geometric series and the formula for calculating the sum to infinity. We've applied the formula correctly and arrived at a consistent result. The fact that our result doesn't match the provided options is a valuable insight in itself.

Conclusion

So, guys, after carefully analyzing the geometric series 8 + 16/3 + 32/9 + ..., we've determined that the sum to infinity is 24. However, this answer isn't present in the given options. It's crucial to remember that sometimes errors occur in problem sets, and it's important to be confident in your calculations and reasoning. We've demonstrated a solid understanding of geometric series and the formula for calculating the sum to infinity. Keep practicing, and you'll become a math whiz in no time! Remember, math isn't just about finding the right answer; it's about understanding the process and thinking critically.