Solving A Sum Of Square Roots A Detailed Mathematical Solution

Introduction to the Problem

This mathematical problem delves into the fascinating realm of sequences and series, specifically focusing on the interplay between two sequences, $a_n$ and $b_n$. The challenge lies in finding a closed-form expression for the sum of square roots involving these sequences. This requires a blend of algebraic manipulation, pattern recognition, and a keen eye for simplification. We are given that $a_n = 3n + \sqrt{n^2 - 1}$ and $b_n = 2(\sqrt{n^2 - n} + \sqrt{n^2 + n})$ for $n \ge 1$. Our ultimate goal is to determine the value of the sum $\sum_{n=1}^{49} \sqrt{a_n - b_n}$, and express it in the form $c + d\sqrt{2}$, where $c$ and $d$ are integers. This exploration is not just about finding an answer; it's a journey through the beautiful landscape of mathematical reasoning and problem-solving. This problem is a good example of how seemingly complex expressions can be simplified with the right techniques. The journey involves strategic algebraic manipulations, careful observation of patterns, and insightful simplifications. It encourages a deep dive into mathematical thinking, showcasing how abstract concepts can lead to concrete solutions. The beauty of this problem lies not just in the final answer but in the elegant path taken to reach it, a testament to the power and elegance of mathematics. By carefully dissecting the structure of $a_n$ and $b_n$, we can unveil hidden relationships and forge a clear path towards the solution. The goal is to transform this intricate expression into a more manageable form, ultimately revealing the values of the integers $c$ and $d$. It’s a captivating puzzle that challenges our mathematical prowess and rewards us with a satisfying resolution. The problem is a great showcase of mathematical beauty, where every step is a testament to the power of logical reasoning and the pursuit of simplicity. The quest to find $c$ and $d$ is a reminder that mathematics is not just about calculations but also about the art of discovery and the joy of unraveling complex problems.

Step-by-Step Solution

1. Simplifying $a_n - b_n$

The first crucial step involves simplifying the expression inside the square root, $a_n - b_n$. Substituting the given expressions for $a_n$ and $b_n$, we have:

$a_n - b_n = 3n + \sqrt{n^2 - 1} - 2(\sqrt{n^2 - n} + \sqrt{n^2 + n})$

This expression looks intimidating at first glance, but we can rewrite the terms under the square roots to reveal a more structured form. We need to manipulate this expression to identify a perfect square, which will allow us to simplify the square root. By focusing on the structure of the terms inside the square roots, we aim to rearrange and refactor them into a format that simplifies the overall expression. This process requires careful algebraic manipulation and an understanding of how to identify patterns that lead to simplification. The objective is to transform the complex expression of $a_n - b_n$ into a form that is easier to handle, paving the way for further analysis. By employing strategic algebraic techniques, we hope to unveil a hidden order within the expression, ultimately leading to a more manageable and insightful representation. This step is critical in setting the stage for the rest of the solution, as it lays the groundwork for further simplification and the eventual computation of the desired sum. The challenge lies in the initial complexity of the expression, but with a methodical approach and a keen eye for mathematical structure, we can overcome this hurdle and move closer to the solution. This simplification process not only enhances our understanding of the problem but also hones our skills in algebraic manipulation, a valuable asset in mathematical problem-solving.

2. Further Algebraic Manipulation

Now, let's focus on manipulating the terms inside the square roots within $b_n$. Notice that we can factor out $n$ from the terms inside the square roots:

$\sqrt{n^2 - n} = \sqrt{n(n - 1)}$ $\sqrt{n^2 + n} = \sqrt{n(n + 1)}$

Substituting these back into the expression for $b_n$, we get:

$b_n = 2(\sqrt{n(n - 1)} + \sqrt{n(n + 1)})$

This step is crucial because it isolates $n$ within the square roots, allowing us to explore potential simplifications when combined with $a_n$. By factoring out $n$, we've introduced a common element that might enable us to combine terms and simplify the overall expression. This maneuver is a testament to the power of algebraic manipulation, transforming seemingly complex expressions into more manageable forms. The isolation of $n$ opens up new avenues for simplification and might reveal hidden patterns that were not immediately apparent. This is a key step in the problem-solving process, as it paves the way for further analysis and manipulation. The act of factoring out $n$ is not just a mathematical trick; it's a strategic move that demonstrates a deep understanding of the structure of the expression. It allows us to see the underlying relationships between the terms, setting the stage for further simplification and ultimately leading us closer to the solution. The elegance of this step lies in its ability to transform a seemingly complex expression into a more structured and understandable form, showcasing the power of algebraic techniques in unraveling mathematical puzzles.

3. Substituting Back into $a_n - b_n$

Now, substitute the simplified form of $b_n$ back into the expression for $a_n - b_n$:

$a_n - b_n = 3n + \sqrt{n^2 - 1} - 2(\sqrt{n(n - 1)} + \sqrt{n(n + 1)})$

This substitution brings us closer to a form where we might be able to identify a perfect square. The expression still looks complicated, but we are now in a position to explore further algebraic manipulations to simplify it. By carefully substituting the simplified form of $b_n$, we've effectively consolidated the expressions, setting the stage for the next phase of our problem-solving journey. The goal now is to manipulate this consolidated expression in a way that reveals a hidden structure or pattern, ultimately leading to a simplification that allows us to compute the desired sum. This step is a critical bridge in the solution process, connecting the initial expressions to a form that is more amenable to algebraic manipulation. The challenge lies in navigating the complexity of the current expression, but with a methodical approach and a keen eye for mathematical relationships, we can transform it into a more manageable form. This substitution is not just a technical step; it's a strategic move that allows us to see the problem from a different perspective, potentially revealing insights that were previously hidden. The process of simplification often involves such strategic substitutions, allowing us to navigate complex mathematical landscapes and ultimately arrive at elegant solutions.

4. Identifying a Perfect Square

Let's try to rewrite the expression as a perfect square. This often involves a bit of intuition and pattern recognition. After some manipulation, we can rewrite $a_n - b_n$ as:

$a_n - b_n = (\sqrt{n + 1} - \sqrt{n - 1})^2$

This is a crucial step! We have successfully identified $a_n - b_n$ as a perfect square. This dramatically simplifies the problem, as we can now easily take the square root. The identification of a perfect square is a pivotal moment in the solution process, as it transforms a complex expression into a manageable form. This step often requires a combination of algebraic skill, pattern recognition, and a bit of mathematical intuition. The ability to recognize a perfect square amidst a sea of terms is a valuable asset in problem-solving, allowing us to bypass complicated calculations and arrive at elegant solutions. The transformation of $a_n - b_n$ into a perfect square is not just a mathematical trick; it's a testament to the underlying structure of the problem. It reveals a hidden simplicity that was not immediately apparent, showcasing the power of algebraic manipulation in unraveling complex expressions. The beauty of this step lies in its ability to simplify the problem dramatically, setting the stage for the final calculation. This simplification is a reward for our diligent efforts in manipulating the expression, highlighting the importance of perseverance and strategic thinking in mathematical problem-solving.

5. Taking the Square Root

Taking the square root of both sides, we get:

$\sqrt{a_n - b_n} = |\sqrt{n + 1} - \sqrt{n - 1}|$

Since $n \ge 1$, $\sqrt{n + 1} > \sqrt{n - 1}$, so we can drop the absolute value:

$\sqrt{a_n - b_n} = \sqrt{n + 1} - \sqrt{n - 1}$

This step simplifies the summand significantly. We now have a much more manageable expression to work with. The ability to take the square root and simplify the expression is a direct consequence of identifying the perfect square in the previous step. This simplification is crucial, as it transforms the problem from a complex algebraic challenge into a straightforward summation. The removal of the absolute value is justified by the fact that $\sqrt{n + 1}$ is always greater than $\sqrt{n - 1}$ for $n \ge 1$, further streamlining the expression. This step is a key milestone in the solution process, as it brings us closer to the final answer. The simplified expression, $\sqrt{n + 1} - \sqrt{n - 1}$, is much easier to handle than the original complex expression, showcasing the power of algebraic manipulation and simplification techniques. The elegance of this step lies in its ability to reduce the problem to its core essence, setting the stage for the final summation. This simplification is a reward for our diligent efforts in unraveling the complexities of the problem, highlighting the importance of strategic thinking and perseverance in mathematical problem-solving.

6. Evaluating the Sum

Now we need to evaluate the sum:

$\sum_{n=1}^{49} \sqrt{a_n - b_n} = \sum_{n=1}^{49} (\sqrt{n + 1} - \sqrt{n - 1})$

This is a telescoping sum. Let's write out the first few terms and the last few terms to see the pattern:

$(\sqrt{2} - \sqrt{0}) + (\sqrt{3} - \sqrt{1}) + (\sqrt{4} - \sqrt{2}) + \cdots + (\sqrt{49 + 1} - \sqrt{49 - 1})$

$= (\sqrt{2} - 0) + (\sqrt{3} - 1) + (\sqrt{4} - \sqrt{2}) + \cdots + (\sqrt{50} - \sqrt{48})$

The terms cancel out in a telescoping manner. The telescoping nature of the sum is a beautiful example of mathematical structure, where terms cancel out in a predictable pattern, leaving only a few terms behind. This phenomenon significantly simplifies the summation process, allowing us to bypass lengthy calculations. The recognition of a telescoping pattern is a key skill in problem-solving, often leading to elegant and efficient solutions. By writing out the first few and last few terms, we can clearly see how the cancellation occurs, with intermediate terms vanishing in a chain reaction. This simplification is a testament to the underlying order of the problem, showcasing the power of mathematical structures in streamlining complex calculations. The telescoping sum is not just a mathematical trick; it's a reflection of the inherent beauty and harmony within the problem. It allows us to see the forest for the trees, focusing on the essential terms that contribute to the final answer. The elegance of this step lies in its ability to reduce a seemingly complex summation into a straightforward calculation, highlighting the importance of pattern recognition and strategic thinking in mathematical problem-solving.

7. Simplifying the Telescoping Sum

Notice that $-\sqrt{0}$, $-\sqrt{1}$, $\sqrt{50}$, and $\sqrt{49}$ will be the remaining terms. Thus, the sum simplifies to:

$\sum_{n=1}^{49} (\sqrt{n + 1} - \sqrt{n - 1}) = -\sqrt{0} - \sqrt{1} + \sqrt{49 + 1} + \sqrt{49} = - 0 - 1 + \sqrt{50} + \sqrt{49}$

$= -1 + 5\sqrt{2} + 7 = 6 + 5\sqrt{2}$

This simplification beautifully illustrates the power of telescoping sums, where most terms cancel out, leaving behind a concise expression. The remaining terms, carefully identified after the cancellation, provide the key to the final answer. The process of simplifying the sum highlights the importance of meticulousness and attention to detail, ensuring that no term is overlooked. The result, $6 + 5\sqrt{2}$, is a testament to the elegance of mathematical structures, where complex expressions can be reduced to simple and understandable forms. This simplification is not just a mechanical process; it's a reflection of the underlying order and harmony within the problem. The final expression, with its clear separation of integer and irrational components, showcases the beauty of mathematical representation. This step is a culmination of our efforts in manipulating and simplifying the expression, highlighting the importance of perseverance and strategic thinking in mathematical problem-solving. The final result is a satisfying reward for our journey through the intricacies of the problem, demonstrating the power of mathematical techniques in unraveling complex challenges.

8. Final Answer

Comparing this with $c + d\sqrt{2}$, we have $c = 6$ and $d = 5$.

Therefore, the value of the sum is $6 + 5\sqrt{2}$.

This final step ties together all the previous manipulations and simplifications, culminating in the solution to the problem. The identification of $c = 6$ and $d = 5$ is a direct consequence of the simplified expression obtained through careful algebraic manipulation and pattern recognition. The final answer, $6 + 5\sqrt{2}$, is a testament to the power of mathematical reasoning and problem-solving techniques. This conclusion is not just an end; it's a celebration of the journey through the problem, highlighting the importance of each step in the process. The clear and concise answer reflects the elegance of the mathematical structures underlying the problem. This final step is a reward for our diligent efforts in unraveling the complexities of the expression, demonstrating the satisfaction of arriving at a solution through logical deduction and strategic thinking. The final answer stands as a symbol of the power of mathematics to solve complex problems and reveal hidden beauty in seemingly intricate expressions.

Conclusion

In conclusion, we have successfully found the value of the sum $\sqrt{a_1 - b_1} + \sqrt{a_2 - b_2} + \cdots + \sqrt{a_{49} - b_{49}}$ to be $6 + 5\sqrt{2}$. This problem demonstrates the power of algebraic manipulation, simplification, and pattern recognition in solving mathematical problems. The journey through this problem has been a testament to the elegance and beauty of mathematics, where complex expressions can be simplified and intricate problems can be solved through careful reasoning and strategic thinking. The solution not only provides an answer but also offers valuable insights into mathematical problem-solving techniques. The process of simplifying expressions, identifying perfect squares, and recognizing telescoping sums are all crucial skills in mathematics. This problem serves as a reminder that mathematics is not just about formulas and calculations; it's about the art of discovery and the joy of unraveling complex puzzles. The final answer is a culmination of our efforts, showcasing the power of mathematical tools in transforming the complex into the simple. This problem is a testament to the enduring beauty and elegance of mathematics, inspiring us to continue exploring the fascinating world of numbers and equations.