Identifying Terms In Arithmetic Sequence A_n = 4n - 7
In the realm of mathematics, sequences hold a fundamental place, providing a structured way to explore patterns and relationships between numbers. Among the various types of sequences, arithmetic sequences stand out for their simplicity and predictability. An arithmetic sequence is characterized by a constant difference between consecutive terms, allowing us to express the sequence using a linear formula. In this comprehensive exploration, we delve into the intricacies of identifying terms within a specific arithmetic sequence defined by the formula a_n = 4n - 7. Our focus will be on determining which numbers from a given set could potentially belong to this sequence. This exploration will not only enhance our understanding of arithmetic sequences but also refine our problem-solving skills in mathematical analysis.
To effectively identify terms within the sequence a_n = 4n - 7, we need to understand the underlying principle: a number can be a term in the sequence if and only if it can be obtained by substituting a positive integer value for n in the formula. This stems from the definition of a sequence, where terms are generated based on a specific order or pattern, and in this case, the pattern is governed by the linear expression 4n - 7. The challenge lies in determining whether a given number, when plugged into the equation 4n - 7 = number, yields a positive integer solution for n. If it does, the number is indeed a term in the sequence; if not, it is excluded. This process involves algebraic manipulation and careful consideration of the constraints imposed by the nature of sequences, particularly the requirement for positive integer indices. The beauty of this approach lies in its systematic nature, allowing us to confidently ascertain the presence or absence of a number within the sequence.
Navigating the process of identifying terms in an arithmetic sequence requires a strategic approach, one that combines algebraic manipulation with a keen understanding of the sequence's properties. We begin by setting up an equation that equates the given number to the sequence's formula, a_n = 4n - 7. This equation serves as the cornerstone of our analysis, allowing us to directly investigate the potential for a positive integer solution for n. Once the equation is established, we embark on the journey of solving for n, employing algebraic techniques such as isolating n on one side of the equation. This often involves adding or subtracting constants and dividing by coefficients, all while maintaining the equation's balance. The solution for n represents the index of the term in the sequence, and its nature – whether it is a positive integer or not – determines whether the given number is a term in the sequence. A positive integer solution signifies that the number is indeed a term, while a non-positive or non-integer solution disqualifies it. This methodical approach ensures accuracy and clarity in our analysis.
In this section, we will methodically analyze each of the given numbers – 20, 21, 28, and 29 – to determine whether they can be terms in the sequence a_n = 4n - 7. For each number, we will follow the procedure outlined earlier: set up the equation, solve for n, and assess the nature of the solution. This step-by-step approach will provide a clear and concise understanding of which numbers belong to the sequence and which do not. This hands-on exploration will solidify our understanding of the principles governing arithmetic sequences and enhance our ability to apply these principles in problem-solving scenarios. The goal is not merely to find the correct answers but also to develop a robust methodology that can be applied to similar problems in the future. Let's embark on this analytical journey, number by number, and unveil the terms that reside within the sequence.
A. Is 20 a Term in the Sequence?
To determine if 20 is a term in the sequence a_n = 4n - 7, we set up the equation 4n - 7 = 20. Our goal is to solve for n and check if the solution is a positive integer. First, we add 7 to both sides of the equation, resulting in 4n = 27. Next, we divide both sides by 4, yielding n = 27/4, which simplifies to n = 6.75. Since 6.75 is not a positive integer, we can confidently conclude that 20 is not a term in the sequence. This methodical approach highlights the importance of obtaining a positive integer solution for n to confirm a number's presence in the sequence. The absence of such a solution indicates that the number does not fit the sequence's pattern. This analysis underscores the crucial role of integer values in defining the terms of a sequence, where each term corresponds to a specific positive integer index.
B. Is 21 a Term in the Sequence?
Now, let's investigate whether 21 can be a term in the sequence a_n = 4n - 7. We initiate the process by setting up the equation 4n - 7 = 21. To isolate n, we first add 7 to both sides of the equation, obtaining 4n = 28. Then, we divide both sides by 4, which gives us n = 7. Since 7 is a positive integer, we can definitively conclude that 21 is a term in the sequence. Specifically, it is the 7th term, as a_7 = 4(7) - 7 = 28 - 7 = 21. This example reinforces the concept that a number is a term in the sequence if and only if the corresponding value of n is a positive integer. The successful identification of 21 as a term showcases the power of algebraic manipulation in dissecting sequences and revealing their constituent elements.
C. Is 28 a Term in the Sequence?
To ascertain whether 28 is a term in the sequence a_n = 4n - 7, we follow the established procedure. We set up the equation 4n - 7 = 28 and proceed to solve for n. Adding 7 to both sides of the equation, we get 4n = 35. Dividing both sides by 4 yields n = 35/4, which simplifies to n = 8.75. Since 8.75 is not a positive integer, we can confidently assert that 28 is not a term in the sequence. This reaffirms the critical role of positive integer solutions in determining membership within a sequence. The fractional value of n indicates that 28 falls between two terms of the sequence but is not itself a member. This analysis further solidifies our understanding of the discrete nature of sequences, where terms are defined at specific integer indices.
D. Is 29 a Term in the Sequence?
Finally, let's examine whether 29 could be a term in the sequence a_n = 4n - 7. We begin by setting up the equation 4n - 7 = 29. To solve for n, we first add 7 to both sides, resulting in 4n = 36. Then, we divide both sides by 4, which gives us n = 9. Since 9 is a positive integer, we can definitively conclude that 29 is a term in the sequence. In fact, it is the 9th term, as a_9 = 4(9) - 7 = 36 - 7 = 29. This final analysis reinforces the fundamental principle that a number is a term in the sequence if and only if the corresponding value of n is a positive integer. The successful identification of 29 as a term solidifies our understanding of the sequence's structure and our ability to apply algebraic techniques to reveal its members.
In this comprehensive exploration, we have successfully navigated the intricacies of identifying terms within the arithmetic sequence a_n = 4n - 7. Through a methodical approach, we have determined that 21 and 29 are terms in the sequence, while 20 and 28 are not. This journey has not only provided specific answers but has also illuminated the fundamental principles governing arithmetic sequences and the techniques for analyzing them. The key takeaway is that a number is a term in the sequence if and only if it corresponds to a positive integer value of n when plugged into the sequence's formula. This understanding forms the bedrock for further explorations in the realm of sequences and series, paving the way for tackling more complex problems and uncovering deeper mathematical insights.
The process of identifying terms in a sequence is not merely a mathematical exercise; it is a gateway to understanding patterns, relationships, and the very fabric of mathematical structures. The ability to dissect a sequence, to discern its members, and to predict its behavior is a powerful tool in the hands of any aspiring mathematician or problem-solver. As we conclude this exploration, let us carry forward the knowledge gained, the techniques honed, and the appreciation for the elegance and precision of mathematics that has been kindled along the way. The world of sequences awaits, brimming with challenges and opportunities for discovery.
