Find The Nth Term Of The Recursive Sequence B(1)=-500, B(n)=b(n-1) * (4/5).
In the fascinating world of mathematics, sequences play a pivotal role, offering a structured way to explore patterns and relationships within numbers. Among the various types of sequences, recursive sequences hold a special place. Recursive sequences are defined by a starting term (or terms) and a recursive formula that relates each term to its preceding term(s). This unique characteristic allows us to build the sequence step-by-step, unveiling the hidden progression. This article focuses on a particular recursive sequence defined by the recurrence relation b(n) = b(n-1) * (4/5) and the initial term b(1) = -500. We will delve into the properties of this sequence, explore its behavior, and uncover the mathematical concepts it exemplifies. This comprehensive examination will provide a solid understanding of how recursive sequences work, their characteristics, and how to analyze them effectively. The journey starts with defining the sequence and progresses to calculating its terms, identifying its type, and finally, deriving its explicit formula, offering a complete picture of this intriguing mathematical entity. Through this exploration, we aim to provide a clear understanding of recursive sequences and their applications in various mathematical contexts. In this specific case, the sequence presents a fascinating scenario where each term is derived from the previous one by multiplying it with a constant factor, hinting towards a geometric progression. This sets the stage for an in-depth analysis of its characteristics and behavior. Understanding the fundamental principles of recursive sequences is crucial not only in mathematics but also in computer science, where they form the basis of many algorithms and data structures. This article aims to provide a solid foundation in this area, equipping readers with the knowledge and skills to tackle more complex problems involving recursive sequences. By breaking down the problem into manageable steps, we will reveal the underlying patterns and relationships within the sequence, making it accessible to readers of all levels.
Defining the Sequence: A Recursive Relationship
Our journey begins by understanding the fundamental definition of the sequence. We are presented with a recursive definition, which means that each term in the sequence is defined in relation to the previous term. The given recursive formula is b(n) = b(n-1) * (4/5), where b(n) represents the nth term of the sequence. This formula tells us that to find any term in the sequence, we need to multiply the previous term by the constant factor 4/5. Additionally, we are provided with the initial term, b(1) = -500, which serves as the starting point for the sequence. Without this initial term, the sequence would be undefined, as we would have no base to begin our calculations. The recursive nature of the sequence highlights the interdependence of its terms. Each term is inextricably linked to its predecessor, creating a chain of values that unfolds as we progress through the sequence. This characteristic distinguishes recursive sequences from explicit sequences, where each term can be calculated independently without reference to previous terms. The recursive definition allows us to generate the sequence iteratively, step by step. Starting with the initial term, we can apply the recursive formula repeatedly to obtain subsequent terms. For instance, to find the second term, b(2), we substitute n = 2 into the formula, yielding b(2) = b(1) * (4/5). Since we know b(1) = -500, we can calculate b(2). Similarly, we can find b(3) by substituting n = 3, and so on. This iterative process reveals the underlying pattern and behavior of the sequence. The recursive relationship also allows us to analyze the long-term behavior of the sequence. By repeatedly applying the formula, we can observe how the terms change as n increases. This can provide insights into the convergence or divergence of the sequence, as well as its overall trend. In the context of our specific sequence, the multiplication by 4/5 suggests a potential decreasing trend, as the factor is less than 1. This observation sets the stage for further investigation into the sequence's behavior and characteristics.
Calculating the First Few Terms: Unveiling the Pattern
To gain a deeper understanding of the sequence, let's calculate the first few terms. This will allow us to observe the pattern and behavior of the sequence more clearly. We already know the first term, b(1) = -500. Using the recursive formula b(n) = b(n-1) * (4/5), we can find the subsequent terms:
- b(2) = b(1) * (4/5) = -500 * (4/5) = -400
- b(3) = b(2) * (4/5) = -400 * (4/5) = -320
- b(4) = b(3) * (4/5) = -320 * (4/5) = -256
- b(5) = b(4) * (4/5) = -256 * (4/5) = -204.8
By calculating these terms, we can observe a clear trend: the terms are decreasing in magnitude, and they are all negative. This is due to the multiplication by the factor 4/5, which is a positive fraction less than 1. Each term is 4/5 of the previous term, leading to a gradual reduction in the absolute value of the terms. The pattern that emerges from these calculations strongly suggests that the sequence is a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant factor, which is known as the common ratio. In our case, the common ratio is 4/5. This observation is crucial because it allows us to apply the properties and formulas associated with geometric sequences to further analyze and understand the behavior of our sequence. Recognizing the sequence as geometric also simplifies the task of finding an explicit formula for the nth term. Instead of relying on the recursive definition, we can use the general formula for a geometric sequence to directly calculate any term without needing to know the previous terms. This is a significant advantage when dealing with large values of n. Furthermore, identifying the sequence as geometric enables us to determine its long-term behavior more easily. We can analyze the common ratio to determine whether the sequence converges, diverges, or oscillates. In this case, since the absolute value of the common ratio is less than 1, we can anticipate that the sequence will converge towards zero as n approaches infinity. This initial exploration of the first few terms provides valuable insights into the nature and behavior of the sequence, paving the way for a more rigorous analysis and understanding.
Identifying the Sequence Type: Geometric Progression
As we observed in the previous section, the sequence exhibits a distinct characteristic: each term is obtained by multiplying the previous term by a constant factor, 4/5. This is the defining property of a geometric sequence. A geometric sequence (also known as a geometric progression) is a sequence where the ratio between consecutive terms remains constant. This constant ratio is called the common ratio, often denoted by 'r'. In our case, the common ratio is 4/5. The recognition of the sequence as geometric is a crucial step in our analysis. Geometric sequences have well-defined properties and formulas that can be applied to simplify calculations and understand their behavior. For instance, the general form of a geometric sequence is given by:
a, ar, ar^2, ar^3, ...
where 'a' is the first term and 'r' is the common ratio. This general form allows us to express any term in the sequence in terms of 'a', 'r', and the term's position in the sequence. The explicit formula for the nth term of a geometric sequence is given by:
b(n) = a * r^(n-1)
where b(n) is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number. This formula is particularly useful because it allows us to calculate any term in the sequence directly, without needing to know the preceding terms. In our specific sequence, the first term is b(1) = -500 and the common ratio is 4/5. Therefore, we can write the explicit formula for our sequence as:
b(n) = -500 * (4/5)^(n-1)
This explicit formula provides a concise and powerful representation of the sequence. It encapsulates the recursive relationship in a single equation, making it easier to analyze and manipulate. Furthermore, recognizing the sequence as geometric allows us to utilize various mathematical tools and techniques that are specifically designed for geometric sequences. For example, we can calculate the sum of a finite number of terms in the sequence using the formula for the sum of a geometric series. We can also analyze the convergence or divergence of the sequence based on the value of the common ratio. In summary, identifying the sequence as a geometric progression is a pivotal step in our analysis. It unlocks a wealth of knowledge and tools that enable us to fully understand and characterize the sequence's behavior.
Deriving the Explicit Formula: A Direct Calculation
Now that we have identified the sequence as geometric and know its first term and common ratio, we can derive the explicit formula for the nth term. As mentioned earlier, the general explicit formula for a geometric sequence is:
b(n) = a * r^(n-1)
where b(n) is the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the term number. In our case, we have a = b(1) = -500 and r = 4/5. Substituting these values into the general formula, we obtain the explicit formula for our sequence:
b(n) = -500 * (4/5)^(n-1)
This formula provides a direct way to calculate any term in the sequence without needing to know the previous terms. For example, if we want to find the 10th term, b(10), we can simply substitute n = 10 into the formula:
b(10) = -500 * (4/5)^(10-1) = -500 * (4/5)^9 ≈ -67.108864
This demonstrates the power and convenience of the explicit formula. It allows us to quickly and efficiently calculate any term in the sequence, regardless of its position. The explicit formula also provides valuable insights into the long-term behavior of the sequence. By analyzing the formula, we can observe that as n increases, the term (4/5)^(n-1) approaches zero because the base (4/5) is a fraction between 0 and 1. Therefore, the terms of the sequence will approach zero as n approaches infinity. This indicates that the sequence converges to zero. The explicit formula can also be used to solve various problems related to the sequence. For example, we can find the term number for which the term's value falls below a certain threshold. We can also analyze the rate at which the sequence converges to zero. The derivation of the explicit formula is a significant achievement in our analysis of the sequence. It provides a complete and concise representation of the sequence, enabling us to perform calculations, analyze its behavior, and solve related problems efficiently. The formula encapsulates the recursive relationship in a direct and manageable form, making it a valuable tool for understanding the sequence.
Conclusion: Summarizing the Sequence's Characteristics
In this comprehensive exploration, we have delved into the intricacies of a recursive sequence defined by b(n) = b(n-1) * (4/5) and b(1) = -500. We began by understanding the recursive definition and its implications. We then calculated the first few terms of the sequence, which revealed a pattern indicative of a geometric progression. This led us to identify the sequence as geometric, with a common ratio of 4/5 and a first term of -500. Subsequently, we derived the explicit formula for the nth term of the sequence: b(n) = -500 * (4/5)^(n-1). This formula provides a direct and efficient way to calculate any term in the sequence. Through our analysis, we have uncovered several key characteristics of the sequence:
- Geometric Nature: The sequence is a geometric progression, where each term is obtained by multiplying the previous term by a constant factor (the common ratio).
- Common Ratio: The common ratio of the sequence is 4/5, which is a positive fraction less than 1.
- Initial Term: The first term of the sequence is -500.
- Explicit Formula: The nth term of the sequence can be calculated directly using the formula b(n) = -500 * (4/5)^(n-1).
- Convergence: The sequence converges to zero as n approaches infinity. This is because the absolute value of the common ratio is less than 1.
- Decreasing Magnitude: The terms of the sequence decrease in magnitude as n increases, approaching zero.
These characteristics provide a complete picture of the sequence's behavior and properties. The explicit formula allows us to make accurate predictions about the sequence's future terms, while the identification of its geometric nature enables us to apply specific mathematical tools and techniques for analysis. The convergence to zero indicates that the terms of the sequence become increasingly smaller as we move further along the sequence. This analysis demonstrates the power of recursive definitions and the importance of identifying patterns in sequences. By understanding the underlying structure of a sequence, we can derive explicit formulas, predict its behavior, and solve related problems effectively. This knowledge is not only valuable in mathematics but also in various fields such as computer science, physics, and engineering, where sequences and patterns play a crucial role. The exploration of this specific sequence serves as a valuable case study for understanding recursive sequences and their applications. It highlights the process of analyzing a recursive definition, identifying its type, deriving an explicit formula, and characterizing its behavior. These skills are essential for tackling more complex problems involving sequences and series.
Recursive sequence, geometric sequence, explicit formula, common ratio, convergence, mathematical analysis, sequence behavior, recursive definition, geometric progression, sequence pattern.