Analyzing Explicit Formulas The Sequences F(n) = 2n + 3 And G(n) = 3(n-1)

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In this article, we delve into the world of sequences and explicit formulas, specifically focusing on two sequences defined by the formulas f(n) = 2n + 3 and g(n) = 3(n-1). Our goal is to analyze these sequences, understand their behavior, and determine the validity of various mathematical statements concerning their terms. We will explore the concepts of explicit formulas, sequence evaluation, and comparative analysis to gain a comprehensive understanding of the relationships between these two sequences. This analysis is crucial for mastering fundamental concepts in mathematics, particularly in algebra and discrete mathematics. By carefully examining the formulas and evaluating specific terms, we can draw meaningful conclusions about the behavior and characteristics of each sequence. This article aims to provide a clear and concise explanation of the process, ensuring that readers can grasp the key concepts and apply them to similar problems.

Understanding Explicit Formulas

To fully grasp the problem, it's essential to first understand what an explicit formula is. An explicit formula provides a direct way to calculate any term in a sequence by substituting the term number (n) into the formula. This contrasts with recursive formulas, which define terms based on preceding terms. In our case, the sequences f(n) and g(n) are both defined using explicit formulas, making it straightforward to find the value of any term without needing to know the previous terms. For instance, to find the 5th term of the sequence f(n), we simply substitute n = 5 into the formula f(n) = 2n + 3. Understanding the nature of explicit formulas is fundamental to working with sequences and series in mathematics. They offer a powerful tool for analyzing patterns and making predictions about the behavior of sequences. By mastering the use of explicit formulas, one can efficiently determine the value of any term in a sequence, which is a crucial skill in various mathematical contexts.

Analyzing the Sequence f(n) = 2n + 3

The sequence f(n) = 2n + 3 is a linear sequence, meaning that the terms increase by a constant difference. To better understand this sequence, let's calculate a few terms. When n = 1, f(1) = 2(1) + 3 = 5. When n = 2, f(2) = 2(2) + 3 = 7. When n = 3, f(3) = 2(3) + 3 = 9. The sequence begins with the terms 5, 7, 9, and so on. Notice that each term is 2 greater than the previous term, indicating a common difference of 2. This constant difference is a key characteristic of linear sequences. The formula itself reveals this linear nature: the term 2n signifies a constant increase proportional to n, and the constant term 3 represents the initial value adjusted by this linear growth. Analyzing such sequences is essential in various mathematical applications, including modeling linear growth patterns and understanding arithmetic progressions. By recognizing the linear nature of the sequence, we can predict its behavior and make informed calculations about its terms.

Analyzing the Sequence g(n) = 3(n - 1)

The sequence g(n) = 3(n - 1) is another linear sequence, but its behavior differs slightly from f(n). Let's calculate a few terms to understand its pattern. When n = 1, g(1) = 3(1 - 1) = 0. When n = 2, g(2) = 3(2 - 1) = 3. When n = 3, g(3) = 3(3 - 1) = 6. The sequence starts with the terms 0, 3, 6, and so on. Here, each term is 3 greater than the previous term, indicating a common difference of 3. The formula g(n) = 3(n - 1) highlights this constant difference and how it influences the sequence's progression. The (n - 1) part shifts the starting point, affecting the sequence's initial terms. Understanding this sequence's linear nature is crucial for recognizing patterns and making accurate predictions about its terms. Like f(n), g(n) can be used in various applications, including modeling linear growth and analyzing arithmetic sequences. By examining the formula and its implications, we gain insights into how the sequence behaves and how its terms relate to each other.

Evaluating the Mathematical Statements

Now that we understand the sequences f(n) and g(n), we can evaluate the given mathematical statements. The statements to consider are:

  • A. g(3) = f(3)
  • B. g(1) > f(1)

To determine which statement is correct, we will substitute the values of n into the respective formulas and compare the results. This process involves careful calculation and a clear understanding of the sequence definitions. By accurately evaluating each expression, we can confidently identify the correct statement and provide a solid justification for our conclusion. This exercise reinforces the importance of precise computation and the ability to interpret mathematical expressions effectively. The following sections will detail the evaluation process for each statement, ensuring a thorough and accurate analysis.

Evaluating Statement A: g(3) = f(3)

To evaluate the statement g(3) = f(3), we need to calculate g(3) and f(3) separately and then compare their values. For f(3), we substitute n = 3 into the formula f(n) = 2n + 3, which gives us f(3) = 2(3) + 3 = 6 + 3 = 9. For g(3), we substitute n = 3 into the formula g(n) = 3(n - 1), which gives us g(3) = 3(3 - 1) = 3(2) = 6. Comparing the results, we see that f(3) = 9 and g(3) = 6. Therefore, the statement g(3) = f(3) is incorrect. This evaluation demonstrates the importance of accurately substituting values into formulas and performing the calculations correctly. By following this process, we can confidently determine the truth or falsehood of mathematical statements and gain a deeper understanding of the relationships between sequences and their terms.

Evaluating Statement B: g(1) > f(1)

To evaluate the statement g(1) > f(1), we need to calculate g(1) and f(1) separately and then compare their values. For f(1), we substitute n = 1 into the formula f(n) = 2n + 3, which gives us f(1) = 2(1) + 3 = 2 + 3 = 5. For g(1), we substitute n = 1 into the formula g(n) = 3(n - 1), which gives us g(1) = 3(1 - 1) = 3(0) = 0. Comparing the results, we see that f(1) = 5 and g(1) = 0. Therefore, the statement g(1) > f(1) is incorrect because 0 is not greater than 5. This evaluation underscores the significance of precise calculations and accurate comparisons when working with mathematical inequalities. By carefully following the steps and substituting values correctly, we can confidently assess the validity of mathematical statements and deepen our understanding of numerical relationships.

Conclusion

In conclusion, after evaluating the mathematical statements concerning the sequences f(n) = 2n + 3 and g(n) = 3(n - 1), we have determined that both statements A (g(3) = f(3)) and B (g(1) > f(1)) are incorrect. This analysis involved calculating specific terms of each sequence by substituting values into their explicit formulas and then comparing the results. This exercise highlights the importance of understanding explicit formulas, performing accurate calculations, and making logical comparisons. By mastering these skills, one can confidently analyze sequences and other mathematical concepts. The process of evaluating mathematical statements is a fundamental aspect of problem-solving in mathematics, requiring careful attention to detail and a thorough understanding of the underlying principles. This article has provided a clear example of how to approach such evaluations and arrive at accurate conclusions.