Finding The Term Position In A Sequence: T(n) = 3n + 41

Understanding Sequences and Terminology

In the world of mathematics, sequences play a crucial role. A sequence is essentially an ordered list of numbers, often following a specific pattern or rule. Each number in the sequence is called a term, and the position of a term within the sequence is denoted by its term number. For instance, the first term is at position 1, the second term is at position 2, and so on. To understand finding the term position, it's essential to grasp the relationship between the term number (n) and the value of the term itself. This relationship is often defined by a formula, allowing us to calculate any term in the sequence if we know its position, or conversely, to find the position of a term if we know its value. Understanding these concepts is fundamental not only for solving problems like the one presented but also for more advanced mathematical topics like series, calculus, and discrete mathematics. The ability to manipulate sequences and understand their underlying structures is a cornerstone of mathematical reasoning and problem-solving. When you encounter a sequence, think of it as a story unfolding, where each term represents a step in the narrative. The formula that defines the sequence is the key to understanding the plot and predicting future events. In our case, the formula T(n) = 3n + 41 is the storyteller, revealing the value of each term based on its position in the sequence. It's a linear relationship, meaning that the terms increase at a constant rate, which simplifies the process of finding a specific term or its position. Mastering sequence analysis is like learning to read the language of numbers, opening up a world of mathematical possibilities and applications.

The Problem: T(n) = 3n + 41 and the Value 359

Our specific problem presents us with a sequence defined by the formula T(n) = 3n + 41. This formula tells us exactly how to calculate any term in the sequence. The T(n) part represents the value of the term at position n. The 3n indicates that each term increases by 3 for every increase in n, and the + 41 is a constant that shifts the entire sequence. This is a classic example of an arithmetic sequence, where the difference between consecutive terms is constant. The problem then states that a particular term in this sequence has a value of 359. Our mission is to find the position (n) of this term within the sequence. This is akin to solving a mystery: we know the outcome (the term's value) and the rule (the formula), and we need to uncover the input (the position). To tackle this, we'll use the power of algebra, a tool that allows us to manipulate equations and isolate the unknown variable. By setting T(n) equal to 359, we create an equation that we can then solve for n. This process demonstrates a fundamental principle in mathematics: the ability to translate a word problem into a mathematical equation, and then to use algebraic techniques to find the solution. This skill is not just useful for solving sequence problems; it's a cornerstone of mathematical literacy, enabling us to model real-world situations and solve complex problems across various disciplines. Understanding how to work with formulas and equations is like having a powerful key that unlocks the doors to mathematical understanding and problem-solving prowess. Embracing algebra opens up a world of possibilities, allowing us to navigate the intricacies of mathematical relationships with confidence and precision.

Solving for n: The Algebraic Approach

Now, let's dive into the core of the problem: finding the position of the term with a value of 359. We start with the given formula, T(n) = 3n + 41, and we know that for a specific term, T(n) = 359. So, we can substitute 359 for T(n) in the equation, giving us 359 = 3n + 41. This is where algebra comes into play. Our goal is to isolate n, which means getting it alone on one side of the equation. The first step is to get rid of the + 41. To do this, we subtract 41 from both sides of the equation. This is a crucial step because it maintains the balance of the equation – whatever we do to one side, we must do to the other. Subtracting 41 from both sides gives us 359 - 41 = 3n + 41 - 41, which simplifies to 318 = 3n. We're now one step closer to finding n. The next step is to get rid of the 3 that's multiplying n. To do this, we divide both sides of the equation by 3. Again, this maintains the balance of the equation. Dividing both sides by 3 gives us 318 / 3 = 3n / 3, which simplifies to 106 = n. And there we have it! We've successfully solved for n. This algebraic process demonstrates the power of manipulating equations to isolate the unknown. It's a technique that's used throughout mathematics and many other fields. Each step, from subtracting 41 to dividing by 3, is a deliberate move, carefully chosen to bring us closer to the solution. Mastering these algebraic techniques is like learning the rules of a game; once you know them, you can play with confidence and skill. The ability to solve equations is a fundamental skill that empowers you to tackle a wide range of mathematical problems with precision and understanding.

The Answer and Its Significance

After carefully applying the algebraic steps, we've arrived at the solution: n = 106. This means that the term with a value of 359 is the 106th term in the sequence defined by T(n) = 3n + 41. This answer is not just a number; it's a piece of information that tells us something specific about the sequence. It tells us where the term 359 fits within the ordered list of numbers. The significance of this answer lies in its connection to the formula T(n) = 3n + 41. We've used the formula both to understand the sequence and to solve for a specific position. This highlights the power of mathematical formulas as tools for understanding and predicting patterns. By understanding the relationship between n and T(n), we can not only find the position of a given term but also calculate the value of any term in the sequence. This ability to work with formulas and equations is a fundamental skill in mathematics and its applications. It's like having a decoder ring that allows you to decipher the secrets hidden within numerical patterns. The answer n = 106 is a testament to the power of algebraic problem-solving and the beauty of mathematical relationships. Understanding the significance of the solution is just as important as finding it, as it allows us to connect the abstract world of mathematics to the concrete world of problem-solving and analysis.

Conclusion: The Power of Mathematical Problem-Solving

In conclusion, we've successfully found the position of the term with a value of 359 in the sequence defined by T(n) = 3n + 41. We achieved this by understanding the nature of sequences, applying algebraic techniques to solve an equation, and interpreting the meaning of the solution. This exercise demonstrates the power of mathematical problem-solving, not just in finding answers but also in developing critical thinking skills. The process of breaking down a problem, identifying the relevant information, choosing the appropriate tools (in this case, algebra), and executing the solution is a valuable skill that transcends mathematics. It's a skill that's applicable to many areas of life, from science and engineering to finance and everyday decision-making. The ability to think logically, to analyze information, and to solve problems is a hallmark of an educated mind. Mathematics, with its emphasis on precision and rigor, is an excellent training ground for these skills. By engaging with mathematical problems, we not only learn specific concepts and techniques but also cultivate a mindset of problem-solving that serves us well in all aspects of life. The journey from the initial problem statement to the final solution is a journey of intellectual discovery, a journey that empowers us to tackle challenges with confidence and competence. Embracing mathematical problem-solving is like sharpening a powerful tool that can be used to shape our understanding of the world and our ability to navigate its complexities.

For further exploration of sequences and mathematical problem-solving, you might find valuable resources at Khan Academy's Algebra I Course.