Step-by-Step Guide Calculating The Sum 1/6 - 1/12 + 1/20 + ... + 1/110
Hey guys! Ever stumbled upon a seemingly complex math problem that looks intimidating at first glance? Well, today we're going to tackle one of those together. We'll break down the calculation of the sum 1/6 - 1/12 + 1/20 + 1/30 + ... + 1/110. Don't worry, it's not as scary as it looks! We'll go through it step-by-step, making sure everyone can follow along. So, grab your thinking caps, and let's dive into this mathematical adventure!
Understanding the Series
Before we jump into the calculations, let's take a closer look at the series itself. Understanding the pattern is key to solving this problem efficiently. The series is 1/6 - 1/12 + 1/20 + 1/30 + ... + 1/110. Notice how the denominators are increasing? They are 6, 12, 20, 30, and so on, up to 110. These numbers aren't just random; they follow a specific pattern. Our main keyword here is series pattern. We can express these denominators as a product of two consecutive numbers. For instance, 6 is 2 * 3, 12 is 3 * 4, 20 is 4 * 5, 30 is 5 * 6, and so forth. Spotting this pattern is a crucial first step because it allows us to rewrite each fraction in a way that simplifies the summation. This rewriting involves using something called partial fraction decomposition, which we'll get into shortly. Recognizing this pattern not only makes the problem solvable but also showcases a common technique used in various mathematical contexts. By identifying the underlying structure, we transform a seemingly complex series into a manageable sequence of operations. Furthermore, understanding the series helps us appreciate the elegance and interconnectedness of mathematical concepts. The ability to recognize patterns is a fundamental skill in mathematics and is applicable in various fields beyond just solving equations. So, let's keep this in mind as we move forward and unravel the mysteries of this sum.
Partial Fraction Decomposition
Okay, now that we've identified the pattern in the denominators, let's talk about partial fraction decomposition. This might sound like a mouthful, but it's a super useful technique for simplifying fractions, especially in series like this one. Partial fraction decomposition is basically a way of breaking down a complex fraction into simpler fractions. In our case, we want to rewrite each term in the series (like 1/6, 1/12, etc.) as the difference of two fractions. Why? Because it will make the summation much easier! Remember how we noticed that the denominators are products of consecutive numbers? This is where that observation becomes really handy. Let's take a general term 1/(n * (n+1)). We can decompose this into the form A/n - B/(n+1). The goal here is to find the values of A and B that make this equation true. To do that, we can combine the fractions on the right side by finding a common denominator, which would be n * (n+1). So, we have [A(n+1) - Bn] / [n(n+1)]. For this to be equal to 1/[n(n+1)], the numerators must be equal. That means A(n+1) - Bn must equal 1. By cleverly choosing values for n (like n=0 and n=-1), we can solve for A and B. You'll find that A = 1 and B = 1. Therefore, 1/[n(n+1)] can be decomposed into 1/n - 1/(n+1). This is the key to simplifying our series. By applying this decomposition to each term, we'll see a lot of cancellation happening, which will lead us to the final answer. So, stay tuned as we put this into action in the next section! This technique is not just a trick; it's a powerful tool in calculus and other areas of mathematics for simplifying complex expressions and making them easier to work with. So, mastering partial fraction decomposition will definitely come in handy in your mathematical journey.
Applying the Decomposition to the Series
Alright, let's get our hands dirty and apply the partial fraction decomposition we just learned to our series. This is where the magic really happens! Our series is 1/6 - 1/12 + 1/20 + 1/30 + ... + 1/110. We've already established that the denominators can be written as products of consecutive numbers: 6 = 2 * 3, 12 = 3 * 4, 20 = 4 * 5, 30 = 5 * 6, and so on. The last term, 1/110, can be written as 1/(10 * 11). Now, using the partial fraction decomposition we figured out earlier, we can rewrite each term in the series as the difference of two fractions. So, 1/6 becomes 1/2 - 1/3, 1/12 becomes 1/3 - 1/4, 1/20 becomes 1/4 - 1/5, and so on. If we write out the first few terms and the last few terms, we get: (1/2 - 1/3) - (1/3 - 1/4) + (1/4 - 1/5) + (1/5 - 1/6) + ... + (1/10 - 1/11). Notice anything cool happening here? A lot of terms are canceling each other out! The -1/3 in the first term cancels with the +1/3 in the second term. Similarly, the -1/4 cancels with the +1/4, and so on. This is called a telescoping series, where intermediate terms cancel out, leaving only the first and last terms. This cancellation is the direct result of our clever use of partial fraction decomposition. It transforms a potentially long and tedious summation into a simple subtraction. The application of decomposition simplifies the whole process. This technique is a testament to the power of mathematical manipulation and how rewriting expressions in different forms can lead to significant simplifications. This telescoping behavior is a common characteristic of series that can be decomposed into partial fractions, making it a valuable tool for solving similar problems. So, let's see what's left after all the cancellation in the next step!
Calculating the Final Sum
Okay, guys, we're in the home stretch now! We've done the hard work of understanding the series, using partial fraction decomposition, and watching the magic of cancellation happen. Now, let's calculate the final sum. Remember how we rewrote the series? It looked like this: (1/2 - 1/3) - (1/3 - 1/4) + (1/4 - 1/5) + (1/5 - 1/6) + ... + (1/10 - 1/11). And we saw how most of the terms canceled each other out. If we carefully track the cancellations, we'll see that the -1/3 cancels with the +1/3, the -1/4 cancels with the +1/4, and so on, all the way up to the -1/10 canceling with a +1/10 somewhere in the middle. So, what are we left with? We're left with the first term, 1/2, and the last term, -1/11. Therefore, the sum of the series is simply 1/2 - 1/11. Now, to subtract these fractions, we need a common denominator. The least common multiple of 2 and 11 is 22. So, we can rewrite 1/2 as 11/22 and 1/11 as 2/22. Now we can subtract: 11/22 - 2/22 = 9/22. And there you have it! The sum of the series 1/6 - 1/12 + 1/20 + 1/30 + ... + 1/110 is 9/22. Isn't that neat? By breaking down a complex problem into smaller, manageable steps, we were able to find the solution. This problem showcases the beauty and power of mathematical techniques like partial fraction decomposition and telescoping series. The final calculation is straightforward after all the simplification. So, the next time you encounter a seemingly daunting math problem, remember this example and think about how you can break it down into simpler steps. You might be surprised at what you can achieve!
Conclusion
So, there you have it! We've successfully calculated the sum of the series 1/6 - 1/12 + 1/20 + 1/30 + ... + 1/110. We started by understanding the pattern in the denominators, then we learned about partial fraction decomposition, applied it to the series, and watched the magic of cancellation unfold. Finally, we performed a simple subtraction to arrive at the answer: 9/22. This problem wasn't just about finding a number; it was about learning a process. We saw how breaking down a complex problem into smaller steps, recognizing patterns, and using appropriate mathematical techniques can make even the most intimidating problems solvable. Remember, math isn't just about memorizing formulas; it's about understanding concepts and developing problem-solving skills. The techniques we used here, like partial fraction decomposition and recognizing telescoping series, are valuable tools that you can apply to other problems as well. The conclusion of our journey highlights the importance of structured problem-solving. Keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is full of fascinating problems just waiting to be solved. And who knows, maybe you'll discover some new mathematical magic along the way! We hope this step-by-step guide has been helpful and has demystified this type of series calculation. Happy mathing, everyone!
