Determining The Sum Of The Series 1/6 + 1/12 + 1/20 + 1/30 + ... + 1/110

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Hey guys! Today, we're diving into a fascinating math problem that involves finding the sum of a series. Specifically, we're going to tackle the series 1/6 + 1/12 + 1/20 + 1/30 + ... + 1/110. This might look intimidating at first, but don't worry, we'll break it down step by step and discover the elegant solution together. Get ready to put on your thinking caps and explore the world of mathematical series!

Understanding the Series: Spotting the Pattern

To even begin calculating the sum of this series, we need to first and foremost understand the series itself. What kind of pattern do the fractions follow? Can we identify a general rule that governs the terms in the series? This is crucial because recognizing the pattern is the key to unlocking the solution. So, let's take a closer look at the denominators: 6, 12, 20, 30, ..., 110. What do these numbers have in common? You might notice that they are not simply increasing by a constant amount, meaning this isn't a straightforward arithmetic sequence. However, a closer inspection reveals something quite interesting. These numbers can be expressed as the product of two consecutive integers.

Let’s break it down further. We can rewrite the denominators as follows:

  • 6 = 2 × 3
  • 12 = 3 × 4
  • 20 = 4 × 5
  • 30 = 5 × 6
  • ...
  • 110 = 10 × 11

Aha! Now the pattern becomes crystal clear. Each denominator is the product of two consecutive integers. This is a significant observation because it allows us to rewrite each term in the series using a technique called partial fraction decomposition. This technique is a powerful tool for simplifying complex fractions and will be instrumental in finding the sum of our series. In essence, partial fraction decomposition allows us to break down a single fraction into a sum or difference of simpler fractions. This will make the summation process much more manageable. By rewriting each term in this way, we'll be able to see a beautiful cancellation pattern emerge, leading us to the final answer. So, keep this pattern in mind as we move on to the next step, where we'll put partial fraction decomposition into action.

The Power of Partial Fraction Decomposition

Now that we've identified the pattern in the denominators, it's time to wield the magic of partial fraction decomposition. This technique is our secret weapon for simplifying the series and making it easier to sum. Remember how we expressed each denominator as the product of two consecutive integers? This is where that comes into play. The core idea behind partial fraction decomposition is to rewrite each fraction in our series as the difference of two simpler fractions. For a general term of the form 1/(n(n+1)), we can decompose it as:

1/(n(n+1)) = 1/n - 1/(n+1)

This might seem a bit abstract, so let's see it in action with our series. Applying this decomposition to the first few terms, we get:

  • 1/6 = 1/(2×3) = 1/2 - 1/3
  • 1/12 = 1/(3×4) = 1/3 - 1/4
  • 1/20 = 1/(4×5) = 1/4 - 1/5
  • 1/30 = 1/(5×6) = 1/5 - 1/6

Do you see the magic happening? Each term is now expressed as the difference of two fractions with consecutive denominators. This is extremely important because it sets the stage for a cascading cancellation effect when we sum the series. This is the heart of the solution, guys! By rewriting the fractions in this way, we've transformed the problem into something much more manageable. We're no longer dealing with a seemingly complex series of fractions; instead, we have a series of differences that are poised to simplify beautifully. This technique highlights the power of mathematical manipulation – by changing the form of the expression, we can reveal its underlying structure and unlock its secrets. In the next step, we'll see this cancellation in action and watch the series collapse to a surprisingly simple result. So, stay tuned and get ready to witness the elegance of mathematics!

The Telescoping Series: A Cascade of Cancellations

With each term beautifully decomposed into the difference of two fractions, we are now ready to witness the magic of a telescoping series. A telescoping series is one where most of the terms cancel out when we add them together, leaving only a few terms at the beginning and the end. This is exactly what we've set up with our partial fraction decomposition. Let's write out the first few terms and the last term of the series in their decomposed form:

(1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + (1/5 - 1/6) + ... + (1/10 - 1/11)

Now, take a very close look. Do you see the pattern? The -1/3 in the first term cancels with the +1/3 in the second term. The -1/4 in the second term cancels with the +1/4 in the third term. This cancellation continues throughout the series, like a chain reaction, or like a telescope collapsing! This is why it's called a telescoping series – the intermediate terms effectively slide into each other and disappear. This is a crucial insight, so make sure you grasp the concept of how these terms are cancelling each other out.

If we were to write out all the terms, we would see that every fraction except the very first (1/2) and the very last (-1/11) gets cancelled out. Therefore, the sum of the series simplifies dramatically to:

1/2 - 1/11

This is a remarkable simplification! What started as a series of seemingly complicated fractions has been reduced to a simple subtraction problem. This highlights the power of recognizing patterns and using appropriate mathematical techniques to simplify complex expressions. The telescoping nature of the series is a beautiful example of mathematical elegance, where a clever manipulation leads to a surprisingly simple result. Now, all that remains is to perform the subtraction and find the final answer. So, let's move on to the last step and wrap up this fascinating problem.

The Grand Finale: Calculating the Sum

After all the clever decomposition and cascading cancellations, we've arrived at the final stage: calculating the sum. We've successfully reduced the original series to a simple subtraction problem:

1/2 - 1/11

To perform this subtraction, we need to find a common denominator for the two fractions. The least common multiple of 2 and 11 is 22. So, we rewrite the fractions with the common denominator:

(1/2) * (11/11) - (1/11) * (2/2) = 11/22 - 2/22

Now, we can easily subtract the numerators:

11/22 - 2/22 = (11 - 2) / 22 = 9/22

And there we have it! The sum of the series 1/6 + 1/12 + 1/20 + 1/30 + ... + 1/110 is 9/22. This is a fantastic result that demonstrates the power of mathematical techniques like partial fraction decomposition and the elegance of telescoping series. It's also a reminder that seemingly complex problems can often be solved by breaking them down into smaller, more manageable steps. So, don't be intimidated by a long series of fractions – with the right approach, you can conquer it! This entire process exemplifies how a seemingly daunting mathematical problem can be elegantly solved by recognizing patterns, applying appropriate techniques, and systematically simplifying the expression. The journey from the initial series to the final answer of 9/22 showcases the beauty and power of mathematical reasoning. We hope you've enjoyed this exploration of series and learned some valuable problem-solving skills along the way.

Therefore, the final answer is:

9/22