Evaluating 1/50 - 1/(50*49) - 1/(49*48) Series An In-Depth Exploration

#mainkeywords Evaluating expressions involving series of fractions can often seem daunting at first glance. However, with careful observation and strategic manipulation, complex-looking expressions can be simplified to reveal elegant solutions. In this article, we will delve into a fascinating expression involving a series of fractions and explore a methodical approach to evaluate it precisely. Our focus will be on the expression: ${ \frac{1}{50} - \frac{1}{50 \cdot 49} - \frac{1}{49 \cdot 48} - \cdots - \frac{1}{2 \cdot 1} }$ This problem exemplifies how recognizing patterns and employing algebraic techniques can lead to a clear and concise solution. To fully grasp the problem, it's essential to break down each component and understand their interplay. The series of fractions may initially appear disconnected, but there's an underlying structure that, once identified, simplifies the entire evaluation process. In this detailed exploration, we'll not only solve the expression but also highlight the fundamental mathematical principles that make this solution possible. Understanding these principles is crucial for tackling similar problems and enhancing your problem-solving skills in mathematics. Let’s embark on this mathematical journey and unravel the mystery behind this intriguing expression. The journey begins with a meticulous examination of the expression’s components, and from there, we'll construct a path to the final solution.

Understanding the Expression

At the heart of #mainkeywords evaluating complex expressions is a thorough understanding of their individual components and the relationships between them. Our expression presents a series of fractions, each with a unique structure that contributes to the overall value. The initial term, ${\frac{1}{50}}$, is straightforward, but the subsequent terms introduce a pattern that requires closer examination. Each term after the first is subtracted and takes the form ${\frac{1}{n \cdot (n-1)}}$, where ${n}$ ranges from 50 down to 2. This pattern is crucial because it suggests a possible strategy for simplification. The denominator of each fraction is the product of two consecutive integers, which hints at the potential for partial fraction decomposition. Partial fraction decomposition is a technique used to break down a complex fraction into simpler fractions, making it easier to manipulate and combine terms. In this case, we can rewrite each term of the form ${\frac{1}{n \cdot (n-1)}}$ as the difference of two fractions. This is a key step in simplifying the expression and revealing its underlying structure. The goal here is to transform the series into a form where terms can cancel out, leading to a more manageable expression. By understanding this foundational concept, we can move forward with the evaluation process with a clear strategy in mind. The structure of the expression is not just a collection of terms; it is a pathway to a simplified and elegant solution. Recognizing this structure is the first step in our mathematical journey. Let's delve deeper into the strategy of partial fraction decomposition to unlock the potential for simplification.

Applying Partial Fraction Decomposition

The power of #mainkeywords partial fraction decomposition lies in its ability to transform complex fractions into simpler, more manageable forms. This technique is particularly useful when dealing with fractions where the denominator can be factored into distinct linear factors, as is the case in our expression. Each term of the form ${\frac{1}{n \cdot (n-1)}}$ can be decomposed into the difference of two fractions. Specifically, we can write: ${ \frac{1}{n(n-1)} = \frac{A}{n-1} + \frac{B}{n} }$ To find the values of ${A}$ and ${B}$, we combine the fractions on the right-hand side and equate the numerators: ${ 1 = A \cdot n + B \cdot (n-1) }$ By choosing appropriate values for ${n}$, we can solve for ${A}$ and ${B}$. Let's set ${n = 1}$: ${ 1 = A \cdot 1 + B \cdot (1-1) \implies A = 1 }$ Now, let's set ${n = 0}$: ${ 1 = A \cdot 0 + B \cdot (0-1) \implies B = -1 }$ Therefore, we have: ${ \frac{1}{n(n-1)} = \frac{1}{n-1} - \frac{1}{n} }$ This decomposition is crucial because it transforms each term in our series into a difference, which will lead to significant cancellations when we sum the terms. Understanding the mechanics of partial fraction decomposition is not just about solving this specific problem; it’s about gaining a powerful tool that can be applied to a wide range of mathematical challenges. The beauty of this technique is in its simplicity and effectiveness. By breaking down complex fractions into simpler components, we unlock the potential for further simplification and ultimately arrive at a solution with clarity and precision. With this powerful tool in hand, we are now ready to apply it to our original expression and witness the transformation unfold.

Transforming the Series

With the #mainkeywords partial fraction decomposition formula ${\frac{1}{n(n-1)} = \frac{1}{n-1} - \frac{1}{n}}$ in hand, we can now transform our original series. The series is given by: ${ \frac{1}{50} - \frac{1}{50 \cdot 49} - \frac{1}{49 \cdot 48} - \cdots - \frac{1}{2 \cdot 1} }$ Applying the decomposition to each term (except the first) allows us to rewrite the series as: ${ \frac{1}{50} - \left( \frac{1}{49} - \frac{1}{50} \right) - \left( \frac{1}{48} - \frac{1}{49} \right) - \cdots - \left( \frac{1}{1} - \frac{1}{2} \right) }$ Distributing the negative signs, we get: ${ \frac{1}{50} - \frac{1}{49} + \frac{1}{50} - \frac{1}{48} + \frac{1}{49} - \cdots - \frac{1}{1} + \frac{1}{2} }$ Notice the pattern of cancellation that emerges. The ${-\frac{1}{49}}$ term cancels with the ${+\frac{1}{49}}$ term, the ${-\frac{1}{48}}$ term will cancel with a ${+\frac{1}{48}}$ term later in the series, and so on. This cascading cancellation is a hallmark of telescoping series, where intermediate terms cancel out, leaving only the first and last terms (or a few terms at the beginning and end). This transformation is a pivotal moment in our evaluation, as it simplifies the series into a form where the final result becomes clear. The key is to carefully observe the pattern and identify which terms cancel out, leaving us with a much simpler expression to evaluate. By understanding this transformation, we can proceed to the final step with confidence and precision. The elegance of this approach lies in its ability to convert a seemingly complex series into a manageable form, revealing the underlying simplicity of the expression. Let’s now move on to the final stage and calculate the result after the cancellations.

Performing the Cancellations and Final Evaluation

As we observed in the previous section, the series undergoes a significant cancellation process. After applying the #mainkeywords partial fraction decomposition and distributing the negative signs, we have: ${ \frac{1}{50} - \frac{1}{49} + \frac{1}{50} - \frac{1}{48} + \frac{1}{49} - \cdots - \frac{1}{1} + \frac{1}{2} }$ The terms cancel out in a telescoping manner. Specifically, ${-\frac{1}{49}}$ cancels with ${+\frac{1}{49}}$, ${-\frac{1}{48}}$ cancels with ${+\frac{1}{48}}$, and so on, until we reach the term ${-\frac{1}{2}}$, which cancels with ${+\frac{1}{2}}$ (which appears from the decomposition of ${-\frac{1}{2 \cdot 1}}$). The remaining terms are: ${ \frac{1}{50} + \frac{1}{50} - 1 }$ Combining the terms, we get: ${ \frac{2}{50} - 1 }$ Simplifying the fraction: ${ \frac{1}{25} - 1 }$ Now, we find a common denominator to combine the terms: ${ \frac{1}{25} - \frac{25}{25} }$ Finally, we subtract: ${ -\frac{24}{25} }$ Thus, the final result of the expression is ${-\frac{24}{25}}$. This elegant solution demonstrates the power of strategic algebraic manipulation and the beauty of telescoping series. The cancellations simplified the expression, leading us to a clear and concise answer. This process not only solves the specific problem but also highlights the general principles of problem-solving in mathematics. The ability to recognize patterns, apply appropriate techniques, and meticulously carry out calculations is essential for mathematical success. This journey through the expression has provided valuable insights into the world of series and fractions, equipping us with the skills to tackle similar challenges in the future. In conclusion, the careful application of partial fraction decomposition and the recognition of telescoping patterns have allowed us to evaluate the expression and arrive at the final result of ${-\frac{24}{25}}$.

Conclusion

In this exploration, we successfully #mainkeywords evaluated a complex expression involving a series of fractions. By applying the technique of partial fraction decomposition, we transformed the series into a telescoping form, which allowed for significant cancellations. This strategic approach simplified the expression and led us to the final result of ${-\frac{24}{25}}$. The key takeaways from this exercise include the importance of recognizing patterns, the power of algebraic manipulation, and the elegance of telescoping series. Partial fraction decomposition is a versatile tool that can be applied to a wide range of mathematical problems, and the concept of telescoping series is a recurring theme in calculus and analysis. This problem-solving journey underscores the value of a methodical approach to mathematics. Breaking down a complex problem into smaller, manageable steps, applying appropriate techniques, and carefully executing calculations are essential skills for mathematical success. Moreover, understanding the underlying principles and concepts allows us to generalize our knowledge and apply it to new and challenging problems. The solution presented here is not just an answer; it is a demonstration of mathematical reasoning and problem-solving prowess. By mastering these techniques, we empower ourselves to tackle a wider range of mathematical challenges with confidence and precision. The ability to transform, simplify, and evaluate expressions is at the heart of mathematical problem-solving, and this article has provided a detailed illustration of these skills in action. As we conclude this exploration, we hope that the insights gained here will serve as a valuable foundation for future mathematical endeavors. The journey through this expression has not only provided a solution but also enriched our understanding of mathematical principles and techniques.