Monotonicity Analysis Of The Sequence (18n - 15) / (n + 14)

Introduction to Monotonicity in Sequences

When discussing monotonicity analysis, understanding sequences is fundamental. Monotonicity, in mathematical terms, refers to the consistent behavior of a sequence, specifically whether it is increasing, decreasing, or constant. A sequence is said to be monotonically increasing if each term is greater than or equal to the preceding term. Conversely, it is monotonically decreasing if each term is less than or equal to the preceding term. Sequences that exhibit either of these behaviors are termed monotonic. Identifying monotonicity is crucial in various mathematical analyses, including determining convergence, finding limits, and understanding the overall behavior of sequences. Understanding monotonicity helps predict how a sequence will behave as n approaches infinity, providing critical insights for further mathematical investigations. For example, a bounded monotonic sequence is guaranteed to converge, a cornerstone concept in real analysis.

To formally define these concepts, consider a sequence denoted by {a**n}. This sequence is monotonically increasing if a**n ≤ a**n+1 for all n in the domain (typically natural numbers). Similarly, it is monotonically decreasing if a**n ≥ a**n+1 for all n. If a sequence strictly increases (a**n < a**n+1), it is strictly increasing, and if it strictly decreases (a**n > a**n+1), it is strictly decreasing. A constant sequence is a trivial case of monotonicity where all terms are equal, and thus, it is both monotonically increasing and decreasing. Monotonicity in sequences can be visualized graphically, where an increasing sequence shows a non-decreasing trend, and a decreasing sequence shows a non-increasing trend. The analysis of monotonicity often involves examining the difference between consecutive terms or, for sequences defined by functions, using calculus to analyze the derivative. In summary, understanding the monotonic properties of a sequence is a powerful tool for mathematical analysis, providing a framework for predicting and explaining sequence behavior.

Defining the Sequence {(18n - 15) / (n + 14)}

The sequence in question is defined by the expression a**n = (18n - 15) / (n + 14), where n represents the term number, typically a natural number (1, 2, 3, ...). This expression provides a rule for generating the terms of the sequence. To better understand the sequence, we can calculate the first few terms by substituting n with different values. For instance, when n = 1, a1 = (18(1) - 15) / (1 + 14) = 3 / 15 = 1/5. When n = 2, a2 = (18(2) - 15) / (2 + 14) = 21 / 16. When n = 3, a3 = (18(3) - 15) / (3 + 14) = 39 / 17. By calculating these initial terms, we can start to observe a potential trend in the sequence's behavior, which is a crucial first step in monotonicity analysis. The general form of the sequence suggests that as n increases, the term a**n will approach a certain limit, which can be inferred by analyzing the dominant terms in the numerator and the denominator.

The expression (18n - 15) / (n + 14) is a rational function, a ratio of two polynomials. In this case, both the numerator and the denominator are linear functions of n. As n grows larger, the constant terms (-15 and 14) become less significant compared to the n terms. This observation suggests that the sequence might approach the ratio of the coefficients of n in the numerator and the denominator, which is 18/1 = 18. However, to rigorously determine the monotonicity, we need to investigate whether the terms are consistently increasing or decreasing. The initial terms we calculated give us some intuition, but a formal approach is necessary to confirm the trend for all n. Analyzing the sequence's definition is therefore essential to understanding its long-term behavior and confirming initial observations. By examining the mathematical structure of the sequence, we can apply techniques such as comparing consecutive terms or using calculus to definitively establish its monotonicity.

Method 1: Analyzing the Difference Between Consecutive Terms

A common method for determining the monotonicity of a sequence involves analyzing the difference between consecutive terms. This approach focuses on evaluating whether a**n+1 - a**n is consistently positive, negative, or zero. If a**n+1 - a**n > 0 for all n, the sequence is strictly increasing. If a**n+1 - a**n < 0 for all n, the sequence is strictly decreasing. If a**n+1 - a**n ≥ 0, the sequence is monotonically increasing, and if a**n+1 - a**n ≤ 0, the sequence is monotonically decreasing. To apply this method to the sequence a**n = (18n - 15) / (n + 14), we first need to find an expression for a**n+1. By substituting n with n + 1, we get a**n+1 = (18(n + 1) - 15) / ((n + 1) + 14) = (18n + 3) / (n + 15).

Next, we compute the difference a**n+1 - a**n:

a**n+1 - a**n = [(18n + 3) / (n + 15)] - [(18n - 15) / (n + 14)]

To simplify this expression, we find a common denominator, which is (n + 15)(n + 14):

a**n+1 - a**n = [(18n + 3)(n + 14) - (18n - 15)(n + 15)] / [(n + 15)(n + 14)]

Expanding the numerator, we get:

a**n+1 - a**n = [18n2 + 252n + 3n + 42 - (18n2 + 270n - 15n - 225)] / [(n + 15)(n + 14)]

Simplifying further:

a**n+1 - a**n = [18n2 + 255n + 42 - 18n2 - 255n + 225] / [(n + 15)(n + 14)]

a**n+1 - a**n = 267 / [(n + 15)(n + 14)]

Since n is a natural number, (n + 15) and (n + 14) are both positive. Thus, the denominator is positive, and the numerator 267 is also positive. Therefore, a**n+1 - a**n > 0 for all n. This result indicates that the sequence {(18n - 15) / (n + 14)} is strictly increasing. Analyzing the difference between terms has proven to be an effective method for determining the monotonicity of the sequence, demonstrating that each term is greater than the previous one. This conclusion is a crucial step in understanding the sequence's behavior and predicting its long-term trend.

Method 2: Analyzing the Function Using Calculus

Another powerful method to analyze the monotonicity of a sequence involves treating the sequence as a function and applying calculus techniques, specifically examining the derivative. This approach is particularly useful when the sequence is defined by a continuous function. For the sequence a**n = (18n - 15) / (n + 14), we can consider the corresponding continuous function f(x) = (18x - 15) / (x + 14), where x is a real number. To determine the monotonicity of f(x), we need to find its derivative, f'(x), and analyze its sign.

Using the quotient rule, which states that if f(x) = g(x) / h(x), then f'(x) = [g'(x)h(x) - g(x)h'(x)] / [h(x)]2, we can compute the derivative of f(x):

Let g(x) = 18x - 15 and h(x) = x + 14. Then, g'(x) = 18 and h'(x) = 1.

Applying the quotient rule:

f'(x) = [18(x + 14) - (18x - 15)(1)] / (x + 14)2

Expanding the numerator:

f'(x) = [18x + 252 - 18x + 15] / (x + 14)2

Simplifying:

f'(x) = 267 / (x + 14)2

Now, we analyze the sign of f'(x). Since (x + 14)2 is always positive for x ≠ -14 (and we are only considering x in the domain of the sequence, i.e., positive integers), and 267 is positive, f'(x) > 0 for all x in the domain. This means that the function f(x) is strictly increasing. Using calculus to analyze the derivative allows us to definitively conclude that the continuous function, and thus the corresponding sequence, is strictly increasing.

The fact that f'(x) > 0 implies that the sequence a**n = (18n - 15) / (n + 14) is also strictly increasing. This is because the derivative represents the instantaneous rate of change of the function, and a positive derivative indicates that the function's values are increasing as x increases. Calculus-based analysis provides a rigorous and efficient way to determine the monotonicity of sequences, particularly those defined by rational functions. This method confirms the conclusion we reached using the difference between consecutive terms, reinforcing the understanding that the sequence is strictly increasing. By examining the sign of the derivative, we gain a clear insight into the sequence's behavior, which is a crucial aspect of mathematical analysis.

Conclusion on the Monotonicity of {(18n - 15) / (n + 14)}

In conclusion, through two distinct methods—analyzing the difference between consecutive terms and using calculus to examine the derivative of the corresponding function—we have definitively established that the sequence a**n = (18n - 15) / (n + 14) is strictly increasing. The first method involved computing the difference a**n+1 - a**n, which simplified to 267 / [(n + 15)(n + 14)]. Since this expression is positive for all natural numbers n, we concluded that each term in the sequence is greater than the preceding term. This direct comparison provides a clear and intuitive understanding of the sequence's increasing nature. Concluding the monotonicity analysis, this method underscores the fundamental definition of an increasing sequence.

The second method utilized calculus by considering the function f(x) = (18x - 15) / (x + 14) and finding its derivative. The derivative f'(x) was calculated to be 267 / (x + 14)2, which is also positive for all relevant x. This result confirmed that the function, and hence the sequence, is strictly increasing. Using calculus offers a powerful analytical tool to verify the sequence's behavior. The consistency between the results obtained from both methods strengthens our conclusion about the monotonicity of the sequence. Understanding that the sequence is strictly increasing is valuable for further analysis, such as determining its convergence and finding its limit. Specifically, since the sequence is increasing and bounded above (as it approaches 18), it is guaranteed to converge. This comprehensive analysis highlights the importance of applying multiple approaches in mathematical investigations to ensure accuracy and deepen understanding. In summary, the sequence {(18n - 15) / (n + 14)} exhibits a clear and consistent increasing trend, making it a prime example of a strictly increasing sequence.