Determining Monotonicity Of Sequence { (18n - 15) / (n + 14) }

Introduction: Exploring Sequence Behavior in Mathematics

In the fascinating realm of mathematics, sequences play a pivotal role in various branches, from calculus to discrete mathematics. Understanding the behavior of sequences, whether they increase, decrease, or fluctuate, is crucial for predicting their long-term trends and applications. Monotonicity, a fundamental property of sequences, characterizes their consistent behavior, providing insights into their convergence and bounds. In this article, we delve into the analysis of the sequence ${ \left\{ \frac{18n - 15}{n + 14} \right\} }$, meticulously examining its monotonicity to determine whether it is increasing, decreasing, or exhibits a more complex pattern. We will explore different methods to approach this problem, providing a comprehensive understanding of the underlying concepts and techniques. The journey into the world of sequences and their monotonicity is not just an academic exercise; it's a gateway to understanding patterns, predicting outcomes, and appreciating the elegance of mathematical structures. We embark on this exploration with the goal of demystifying the behavior of sequences, making it accessible and engaging for anyone with a curious mind. Understanding whether a sequence consistently increases or decreases has significant implications in various fields, including computer science, engineering, and economics, where sequences model real-world phenomena. By mastering the techniques to analyze monotonicity, we equip ourselves with a powerful tool for problem-solving and decision-making in a wide range of contexts. This article serves as a comprehensive guide, walking you through the process step by step, ensuring that you grasp the concepts thoroughly and can apply them confidently to other sequences you encounter. Let's embark on this exciting journey of mathematical discovery together!

Defining Monotonicity: Increasing, Decreasing, or Neither

Before we dive into the specifics of the given sequence, let's establish a clear understanding of monotonicity. A sequence is said to be monotonic increasing if each term is greater than or equal to the preceding term. Conversely, a sequence is monotonic decreasing if each term is less than or equal to the preceding term. If a sequence exhibits either of these behaviors, we classify it as monotonic. However, if a sequence fluctuates, sometimes increasing and sometimes decreasing, it is considered non-monotonic. To formally define these concepts, let's consider a sequence denoted by ${ a_n }$, where ${ n }$ represents the term number. The sequence ${ a_n }$ is monotonic increasing if ${ a_{n+1} \geq a_n }$ for all ${ n }$. Similarly, the sequence ${ a_n }$ is monotonic decreasing if ${ a_{n+1} \leq a_n }$ for all ${ n }$. These inequalities provide the mathematical foundation for determining the monotonicity of a sequence. To apply these definitions in practice, we often examine the difference between consecutive terms, ${ a_{n+1} - a_n }$. If this difference is always non-negative, the sequence is increasing. If it's always non-positive, the sequence is decreasing. This approach provides a direct way to assess the sequence's behavior. Understanding the concept of monotonicity is crucial not only for classifying sequences but also for predicting their long-term behavior. For instance, a monotonic bounded sequence is guaranteed to converge, a fundamental result in real analysis. This connection between monotonicity and convergence highlights the importance of this property in mathematical analysis. Moreover, monotonicity plays a vital role in optimization problems, where algorithms often rely on the monotonic behavior of functions to find optimal solutions. In essence, monotonicity serves as a cornerstone concept in mathematics, providing insights into the behavior of sequences and functions, and enabling us to solve a wide range of problems.

Analyzing the Sequence: Method 1 - Examining the Difference

To determine the monotonicity of the sequence ${ \left\{ \frac{18n - 15}{n + 14} \right\} }$, a direct approach is to examine the difference between consecutive terms. Let's denote the ${ n }$-th term of the sequence as ${ a_n = \frac{18n - 15}{n + 14} }$. Our goal is to analyze the sign of the difference ${ a_{n+1} - a_n }$. If ${ a_{n+1} - a_n > 0 }$ for all ${ n }$, the sequence is increasing. If ${ a_{n+1} - a_n < 0 }$ for all ${ n }$, the sequence is decreasing. Let's calculate ${ a_{n+1} }$ by substituting ${ n + 1 }$ for ${ n }$ in the expression for ${ a_n }$: ${ a_{n+1} = \frac{18(n + 1) - 15}{(n + 1) + 14} = \frac{18n + 3}{n + 15} }$. Now, we compute the difference ${ a_{n+1} - a_n }$:

${ a_{n+1} - a_n = \frac{18n + 3}{n + 15} - \frac{18n - 15}{n + 14} }$

To simplify this expression, we find a common denominator:

${ a_{n+1} - a_n = \frac{(18n + 3)(n + 14) - (18n - 15)(n + 15)}{(n + 15)(n + 14)} }$

Expanding the numerators:

${ a_{n+1} - a_n = \frac{(18n^2 + 255n + 42) - (18n^2 + 255n - 225)}{(n + 15)(n + 14)} }$

Simplifying the numerator:

${ a_{n+1} - a_n = \frac{267}{(n + 15)(n + 14)} }$

Since ${ n }$ is a positive integer, the denominator ${ (n + 15)(n + 14) }$ is always positive. The numerator, 267, is also positive. Therefore, the difference ${ a_{n+1} - a_n }$ is always positive for all ${ n }$. This implies that the sequence is monotonic increasing. This method provides a rigorous proof of the sequence's behavior, relying on algebraic manipulation and the definition of monotonicity. By carefully examining the difference between consecutive terms, we have successfully determined that the sequence consistently increases as ${ n }$ increases.

Method 2 - Analyzing the Function: Calculus Approach

Another powerful approach to determine the monotonicity of a sequence involves treating the sequence as a function and applying calculus techniques. Consider the function ${ f(x) = \frac{18x - 15}{x + 14} }$, where ${ x }$ is a real variable. If we can show that the derivative of this function, ${ f'(x) }$, is positive for all ${ x }$ greater than or equal to 1 (since ${ n }$ represents positive integers), then the function is increasing, and consequently, the sequence is also increasing. Let's compute the derivative ${ f'(x) }$ using the quotient rule: ${ f'(x) = \frac{u'v - uv'}{v^2} }$, where ${ u = 18x - 15 }$ and ${ v = x + 14 }$. We have ${ u' = 18 }$ and ${ v' = 1 }$. Applying the quotient rule:

${ f'(x) = \frac{18(x + 14) - (18x - 15)(1)}{(x + 14)^2} }$

Simplifying the numerator:

${ f'(x) = \frac{18x + 252 - 18x + 15}{(x + 14)^2} }$

${ f'(x) = \frac{267}{(x + 14)^2} }$

Since the numerator, 267, is positive and the denominator, ${ (x + 14)^2 }$, is always positive for any real number ${ x }$ (except for ${ x = -14 }$, which is not in our domain), the derivative ${ f'(x) }$ is positive for all ${ x }$ in our domain (${ x \geq 1 }$). This indicates that the function ${ f(x) }$ is monotonic increasing for ${ x \geq 1 }$. Therefore, the sequence ${ \left\{ \frac{18n - 15}{n + 14} \right\} }$ is also monotonic increasing. This calculus-based approach provides an elegant and efficient way to analyze the monotonicity of the sequence. By leveraging the power of derivatives, we can gain insights into the function's behavior and, consequently, the sequence's behavior. This method demonstrates the interconnectedness of calculus and sequence analysis, highlighting the versatility of mathematical tools.

Conclusion: The Sequence is Monotonic Increasing

In conclusion, through two distinct methods – examining the difference between consecutive terms and analyzing the function using calculus – we have definitively established that the sequence ${ \left\{ \frac{18n - 15}{n + 14} \right\} }$ is monotonic increasing. The first method, involving algebraic manipulation, directly compared consecutive terms and revealed a positive difference, confirming the increasing nature of the sequence. The second method, employing calculus, calculated the derivative of the corresponding function and demonstrated its positivity, providing further evidence of the sequence's increasing behavior. These two approaches, while different in their techniques, converge on the same conclusion, reinforcing the robustness of our analysis. Understanding the monotonicity of a sequence is not just an academic exercise; it has practical implications in various fields. For instance, in optimization problems, knowing that a sequence is increasing or decreasing can help us design efficient algorithms to find optimal solutions. In financial modeling, monotonic sequences can represent trends in stock prices or economic indicators, providing valuable insights for decision-making. Moreover, the concept of monotonicity is fundamental to many theorems in calculus and analysis, such as the Monotone Convergence Theorem, which states that a bounded monotonic sequence converges. This theorem highlights the importance of monotonicity in determining the convergence of sequences, a crucial concept in mathematical analysis. By mastering the techniques to analyze monotonicity, we equip ourselves with a powerful tool for understanding and predicting the behavior of sequences, and for solving a wide range of problems in mathematics and its applications. This article has provided a comprehensive guide to analyzing the monotonicity of a specific sequence, but the principles and methods discussed can be applied to a vast array of sequences, empowering you to explore the fascinating world of sequences and their behaviors.

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