Solving Fraction Expressions A Step-by-Step Guide

Hey guys! Today, we are going to tackle a fun mathematical problem that involves calculating the value of an expression with fractions. Specifically, we'll be working with a series of fractions that have a unique pattern in their denominators. So, let's dive right in and break down this problem step by step!

Understanding the Problem

The expression we need to evaluate is:

$\frac{1}{3 \cdot 7}+\frac{1}{7 \cdot 11}+\frac{1}{11 \cdot 15}+\frac{1}{15 \cdot 19}+\frac{1}{19 \cdot 23}$

At first glance, this might seem a bit intimidating, but don't worry! We'll use a clever trick to simplify it. Notice that the denominators are products of two numbers, and there's a consistent difference of 4 between the numbers in each product (7-3 = 4, 11-7 = 4, and so on). This pattern is the key to solving this problem efficiently.

The core concept here is to decompose each fraction into a difference of two simpler fractions. This technique, often called partial fraction decomposition, allows us to rewrite the expression in a way that many terms will cancel out, leaving us with a much simpler calculation. This is the most important step to understand, so let's get into the nitty-gritty of how this works. The essence of this method lies in recognizing the pattern in the denominators. The common difference of 4 is the magic number that will allow us to split each fraction. We'll see how this difference plays out in the decomposition process. This is crucial for those who want to solve similar problems in the future, so pay close attention. Moreover, this method isn't just a one-trick pony; it's a fundamental concept in calculus and other advanced math fields. Understanding it now will make your life a lot easier down the road. So, stick with me, and we'll conquer this problem together! We're going to transform what looks like a complex sum into a straightforward calculation using this powerful technique. The beauty of mathematics lies in these kinds of transformations, where a clever trick can turn a seemingly difficult problem into something manageable. Keep this in mind as we move forward, and you'll start seeing math problems in a whole new light.

Applying Partial Fraction Decomposition

The trick is to rewrite each fraction in the form:

$\frac{1}{n(n+4)} = \frac{A}{n} - \frac{B}{n+4}$

where A and B are constants that we need to determine. However, in this specific case, we can use a shortcut. Since the difference between the factors in the denominator is 4, we aim to express each fraction as a difference with a factor of $\frac{1}{4}$:

$\frac{1}{3 \cdot 7} = \frac{1}{4} \left( \frac{1}{3} - \frac{1}{7} \right)$

Let's verify this. If we combine the fractions inside the parentheses, we get:

$\frac{1}{3} - \frac{1}{7} = \frac{7 - 3}{3 \cdot 7} = \frac{4}{3 \cdot 7}$

Multiplying by $\frac{1}{4}$, we indeed get $\frac{1}{3 \cdot 7}$.

We can apply this same logic to all the fractions in the expression:

• $\frac{1}{7 \cdot 11} = \frac{1}{4} \left( \frac{1}{7} - \frac{1}{11} \right)$
• $\frac{1}{11 \cdot 15} = \frac{1}{4} \left( \frac{1}{11} - \frac{1}{15} \right)$
• $\frac{1}{15 \cdot 19} = \frac{1}{4} \left( \frac{1}{15} - \frac{1}{19} \right)$
• $\frac{1}{19 \cdot 23} = \frac{1}{4} \left( \frac{1}{19} - \frac{1}{23} \right)$

This decomposition is the linchpin of the solution. It transforms the original problem into a telescoping series, where most of the terms will cancel out beautifully. Think of it like a set of dominoes falling; each term knocks out the next, simplifying the whole process. This partial fraction decomposition is a technique that's widely used in calculus for integration, and it's also incredibly useful in discrete mathematics, as we're seeing here. By rewriting each fraction as a difference, we're setting up a cascade of cancellations that will make the final calculation much easier. The key takeaway here is the ability to recognize patterns and apply the right technique. In this case, the consistent difference of 4 in the denominators hinted at the possibility of using this decomposition method. This is the kind of mathematical intuition that develops with practice, so don't be discouraged if it doesn't come naturally at first. Keep solving problems, and you'll start to see these patterns more readily. Remember, mathematics is not just about formulas; it's about recognizing structures and relationships. This step beautifully illustrates that point. By applying this decomposition, we're not just crunching numbers; we're revealing a hidden structure within the problem that makes it much more tractable. So, let's move on to the next step and see how this cascading cancellation unfolds!

Simplifying the Expression

Now, let's substitute these decompositions back into the original expression:

$\frac{1}{3 \cdot 7}+\frac{1}{7 \cdot 11}+\frac{1}{11 \cdot 15}+\frac{1}{15 \cdot 19}+\frac{1}{19 \cdot 23} = \frac{1}{4} \left( \frac{1}{3} - \frac{1}{7} \right) + \frac{1}{4} \left( \frac{1}{7} - \frac{1}{11} \right) + \frac{1}{4} \left( \frac{1}{11} - \frac{1}{15} \right) + \frac{1}{4} \left( \frac{1}{15} - \frac{1}{19} \right) + \frac{1}{4} \left( \frac{1}{19} - \frac{1}{23} \right)$

We can factor out the $\frac{1}{4}$:

$= \frac{1}{4} \left( \frac{1}{3} - \frac{1}{7} + \frac{1}{7} - \frac{1}{11} + \frac{1}{11} - \frac{1}{15} + \frac{1}{15} - \frac{1}{19} + \frac{1}{19} - \frac{1}{23} \right)$

Notice the beautiful cancellation! The $-\frac{1}{7}$ cancels with the $+\frac{1}{7}$, the $-\frac{1}{11}$ cancels with the $+\frac{1}{11}$, and so on. This is called a telescoping series, where most of the terms collapse, leaving only the first and last terms:

$= \frac{1}{4} \left( \frac{1}{3} - \frac{1}{23} \right)$

This cancellation is the heart of the solution's elegance. It dramatically simplifies the expression, transforming what initially looked like a complex sum into a straightforward subtraction problem. The telescoping nature of the series is a direct result of the partial fraction decomposition we applied earlier. This highlights the power of choosing the right technique to solve a mathematical problem. The ability to recognize patterns and apply appropriate methods is a crucial skill in mathematics. This step also demonstrates the importance of paying attention to details. The cancellations happen because of the precise way we decomposed the fractions. If we had made a mistake in that step, the cancellations wouldn't occur, and we'd be stuck with a much more complicated calculation. This telescoping series is a beautiful example of mathematical harmony. It's like a perfectly balanced equation, where every term plays a role in the final result. This is one of the reasons why mathematicians find this kind of problem so satisfying to solve. It's not just about getting the right answer; it's about appreciating the elegance and structure of the mathematics itself. So, let's move on to the final calculation and see how this simplified expression leads us to the solution!

Final Calculation

Now we just need to perform the subtraction and multiply by $\frac{1}{4}$:

$\frac{1}{4} \left( \frac{1}{3} - \frac{1}{23} \right) = \frac{1}{4} \left( \frac{23 - 3}{3 \cdot 23} \right) = \frac{1}{4} \left( \frac{20}{69} \right) = \frac{5}{69}$

So, the value of the expression is $\frac{5}{69}$.

This final calculation is the culmination of all our efforts. It's where we bring together all the pieces of the puzzle and arrive at the solution. The subtraction of the fractions requires finding a common denominator, which in this case is the product of 3 and 23. The arithmetic is straightforward, and we quickly arrive at the simplified fraction $\frac{20}{69}$. The final multiplication by $\frac{1}{4}$ is the last step, and it leads us to the answer: $\frac{5}{69}$. This result confirms the elegance of the method we used. The complex initial expression has been reduced to a simple fraction, thanks to the clever application of partial fraction decomposition and the resulting telescoping series. This problem is a great example of how a strategic approach can make a seemingly difficult task manageable. The key takeaways from this solution are the importance of recognizing patterns, applying appropriate techniques, and paying attention to details. Mathematics is not just about memorizing formulas; it's about developing problem-solving skills and appreciating the beauty of mathematical structures. This problem has given us a glimpse of that beauty, and hopefully, it has inspired you to explore more mathematical challenges. So, let's celebrate our success in solving this problem and move on to new adventures in the world of mathematics!

Conclusion

Therefore, the correct answer is B) $\frac{5}{69}$. We successfully calculated the value of the expression by using partial fraction decomposition and recognizing the telescoping series pattern. Great job, guys!

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