Identifying Sequences Defined By Recursive Formulas A Detailed Explanation

Introduction to Recursive Formulas

In the realm of mathematics, recursive formulas play a pivotal role in defining sequences. A recursive formula is a mathematical expression that defines a sequence by relating each term to the preceding term or terms. This method of definition is particularly useful for sequences where there is a consistent pattern or relationship between consecutive terms. Understanding recursive formulas is crucial for various mathematical applications, including computer science, finance, and physics. To truly grasp the concept, it's essential to break down the components and how they interact. At its core, a recursive formula provides a rule or set of rules that describe how to generate subsequent terms in a sequence based on the values of the preceding terms. This contrasts with explicit formulas, which directly define a term based on its position in the sequence without referencing previous terms. The beauty of recursive formulas lies in their ability to capture the dynamic nature of sequences where each term builds upon the last, creating intricate patterns and relationships. The process of using a recursive formula typically begins with one or more initial terms, which serve as the foundation for generating the rest of the sequence. These initial terms are essential because they provide the starting point for the recursive process. Without them, the formula would be unable to produce any terms, as it relies on the values of previous terms to calculate the next. Once the initial terms are established, the recursive formula is applied repeatedly to generate subsequent terms. Each application of the formula takes the previous term (or terms) as input and produces the next term in the sequence. This iterative process continues, with each new term being calculated based on the ones that came before, until the desired number of terms has been generated or a specific condition is met.

Understanding the Given Recursive Formula: f(n+1) = f(n) + 2.5

Let's delve into the specifics of the recursive formula provided: f(n+1) = f(n) + 2.5. This formula tells us a great deal about the sequence it defines. The formula f(n+1) = f(n) + 2.5 is a classic example of a linear recurrence relation, where each term is obtained by adding a constant value to the previous term. This type of formula defines an arithmetic sequence, characterized by a constant difference between consecutive terms. In this particular case, the constant difference is 2.5, which means that each term in the sequence will be 2.5 greater than the term before it. This constant difference is the hallmark of arithmetic sequences and distinguishes them from other types of sequences, such as geometric sequences, where the terms are related by a constant ratio rather than a constant difference. The notation used in the formula, f(n+1) and f(n), is crucial for understanding the recursive nature of the sequence. The term f(n) represents the nth term in the sequence, while f(n+1) represents the term that comes immediately after it. The formula states that to find the (n+1)th term, you simply add 2.5 to the nth term. This iterative process is what defines the sequence and allows it to be generated term by term. To fully understand how this formula works, it's helpful to consider an example. Suppose we have the first term of the sequence, f(1), which is equal to 1. Using the formula, we can find the second term, f(2), by adding 2.5 to f(1), which gives us 1 + 2.5 = 3.5. Similarly, to find the third term, f(3), we add 2.5 to f(2), resulting in 3.5 + 2.5 = 6. This process can be repeated indefinitely to generate as many terms of the sequence as desired. The formula f(n+1) = f(n) + 2.5 effectively describes a sequence where each term is obtained by adding 2.5 to the previous term. This constant addition creates a linear progression, making the sequence an arithmetic sequence with a common difference of 2.5. Understanding this formula is key to identifying which sequences could be partially defined by it.

Analyzing the Given Sequences

Now, let's apply our understanding of the recursive formula to analyze the given sequences and determine which one could be partially defined by f(n+1) = f(n) + 2.5. Each sequence presents a unique set of numbers, and our task is to identify the sequence where the difference between consecutive terms consistently matches the +2.5 rule. To do this effectively, we will systematically examine each sequence, calculate the differences between successive terms, and compare these differences to the 2.5 constant specified in the recursive formula. This process will allow us to determine whether each sequence adheres to the arithmetic progression defined by the formula or whether it follows a different pattern. The first sequence we'll consider is A: 2.5, 6.25, 15.625, 39.0625, .... To analyze this sequence, we calculate the differences between consecutive terms. The difference between 6.25 and 2.5 is 3.75, and the difference between 15.625 and 6.25 is 9.375. Since the differences are not constant and do not equal 2.5, this sequence does not conform to the given recursive formula. Next, we examine sequence B: 2.5, 5, 10, 20, .... Again, we calculate the differences between successive terms. The difference between 5 and 2.5 is 2.5, but the difference between 10 and 5 is 5, and the difference between 20 and 10 is 10. As the differences are not constant, this sequence also does not align with the recursive formula f(n+1) = f(n) + 2.5. Finally, we analyze sequence C: -10, -7.5, -5, -2.5, .... Calculating the differences, we find that the difference between -7.5 and -10 is 2.5, the difference between -5 and -7.5 is 2.5, and the difference between -2.5 and -5 is also 2.5. Since the difference between each consecutive term is consistently 2.5, this sequence satisfies the condition specified by the recursive formula f(n+1) = f(n) + 2.5. By systematically analyzing each sequence and calculating the differences between consecutive terms, we have determined that only sequence C adheres to the rule defined by the recursive formula f(n+1) = f(n) + 2.5.

Determining the Correct Sequence

After meticulously analyzing each sequence, we've arrived at a definitive conclusion. Sequence A exhibits differences between terms that are not consistent with the +2.5 increment dictated by the recursive formula f(n+1) = f(n) + 2.5. The varying differences disqualify it from being defined by this particular recursive relationship. Likewise, sequence B also fails to meet the criteria. The differences between its terms fluctuate, diverging from the constant 2.5 addition required by the formula. This inconsistency eliminates sequence B as a potential match for the given recursive definition. However, sequence C presents a different scenario. Upon examination, it reveals a consistent pattern where each term is precisely 2.5 greater than its predecessor. This constant difference aligns perfectly with the recursive formula f(n+1) = f(n) + 2.5, making sequence C a viable candidate. Therefore, based on our analysis, sequence C is the sequence that could be partially defined by the recursive formula f(n+1) = f(n) + 2.5. The consistent addition of 2.5 between successive terms is the key characteristic that confirms this conclusion. In contrast, sequences A and B display variable differences, indicating that they follow different patterns or recursive relationships. This systematic evaluation underscores the importance of understanding the underlying principles of recursive formulas and how they govern the generation of sequences. By applying this knowledge, we can effectively analyze and identify sequences that adhere to specific recursive definitions.

Conclusion: Sequence C is the Answer

In conclusion, after a thorough examination of the given sequences and their relationship to the recursive formula f(n+1) = f(n) + 2.5, we have definitively identified sequence C as the correct answer. Sequence C, represented by -10, -7.5, -5, -2.5, ..., stands out because it consistently adheres to the rule specified by the recursive formula. Each term in sequence C is precisely 2.5 greater than the preceding term, aligning perfectly with the f(n+1) = f(n) + 2.5 relationship. This consistent addition of 2.5 is the hallmark of an arithmetic sequence, which is precisely the type of sequence defined by the given recursive formula. This adherence to the recursive rule makes sequence C the only viable option among the provided choices. In contrast, sequences A and B deviate from this pattern. Sequence A exhibits differences between terms that vary, indicating a non-constant relationship between successive elements. Similarly, sequence B displays inconsistent differences, further disqualifying it from being defined by the recursive formula f(n+1) = f(n) + 2.5. The systematic analysis of each sequence highlights the importance of recognizing and applying the principles of recursive formulas. By understanding how these formulas define relationships between terms in a sequence, we can effectively identify sequences that conform to specific recursive definitions. The consistent difference of 2.5 observed in sequence C serves as a clear indicator of its alignment with the recursive formula, reinforcing our conclusion that sequence C is the answer. This exercise demonstrates the power of recursive formulas in defining and analyzing sequences, providing a valuable tool for mathematical exploration and problem-solving.