Identifying Functions For Arithmetic Sequence 8, 1.5, -5, -11.5

Understanding arithmetic sequences is fundamental in mathematics, particularly in algebra and calculus. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference. Identifying the functions that accurately represent a given arithmetic sequence is a crucial skill, as it allows us to predict future terms and understand the sequence's underlying pattern. In this article, we will delve into the arithmetic sequence 8, 1.5, -5, -11.5... and explore the various functions that correctly represent it. This will involve examining both explicit formulas and recursive definitions, providing a comprehensive understanding of how to express arithmetic sequences mathematically.

The given sequence, 8, 1.5, -5, -11.5..., presents a classic example of an arithmetic sequence. To determine the functions that accurately represent this sequence, we first need to identify the common difference. By subtracting consecutive terms, we find that 1.5 - 8 = -6.5, -5 - 1.5 = -6.5, and -11.5 - (-5) = -6.5. Thus, the common difference is -6.5. This common difference is the cornerstone for building both explicit and recursive formulas that describe the sequence. Explicit formulas allow us to calculate any term in the sequence directly, while recursive formulas define a term based on the preceding term. Understanding both types of formulas is essential for a complete grasp of arithmetic sequences. We will explore how each function provided either captures or fails to capture this common difference and the initial term, which is also vital for accurately representing the sequence.

In the following sections, we will dissect the provided functions, assessing their ability to generate the given sequence. We will start by examining the explicit formulas, which provide a direct relationship between the term number (n) and the term value. These formulas are typically in the form f(n) = an + b, where a represents the common difference and b is a constant related to the initial term. We will then move on to the recursive definitions, which define the sequence by specifying the first term and a rule for obtaining subsequent terms. Recursive definitions are typically in the form f(1) = c, f(n+1) = f(n) + d, where c is the first term and d is the common difference. By carefully analyzing each function, we can determine which ones accurately represent the arithmetic sequence 8, 1.5, -5, -11.5..., ensuring a solid understanding of arithmetic sequence representation.

Analyzing the Arithmetic Sequence 8, 1.5, -5, -11.5…

To accurately represent the arithmetic sequence 8, 1.5, -5, -11.5…, we must first dissect its components and identify key characteristics. As previously established, the common difference is -6.5. This means that each term in the sequence is obtained by subtracting 6.5 from the preceding term. The first term of the sequence, often denoted as f(1), is 8. These two pieces of information – the common difference and the first term – are critical for constructing both explicit and recursive formulas that define the sequence. The common difference dictates the slope of the linear function that represents the sequence, while the first term anchors the sequence to a specific starting point.

Understanding these fundamental aspects allows us to evaluate the given functions effectively. An explicit formula for an arithmetic sequence typically takes the form f(n) = a + (n - 1)d, where a is the first term, d is the common difference, and n is the term number. In our case, a = 8 and d = -6.5. Therefore, the explicit formula for this sequence should be a variation of f(n) = 8 + (n - 1)(-6.5). Simplifying this formula will give us a standard form that we can compare with the provided functions. This comparison will reveal which explicit formulas correctly capture the relationship between the term number and the term value in the sequence. It is crucial to ensure that the coefficient of n matches the common difference and that the constant term aligns with the adjusted initial term.

Furthermore, a recursive definition for an arithmetic sequence consists of two parts: specifying the first term and providing a rule for finding subsequent terms. The general form of a recursive definition is f(1) = a, f(n+1) = f(n) + d, where a is the first term and d is the common difference. For our sequence, this translates to f(1) = 8 and f(n+1) = f(n) - 6.5. This recursive definition states that the first term is 8, and each subsequent term is obtained by subtracting 6.5 from the previous term. When evaluating the provided recursive functions, we must verify that both the initial condition (f(1) = 8) and the recursive step (f(n+1) = f(n) - 6.5) are accurately represented. Any deviation from these conditions will result in a function that does not correctly describe the given arithmetic sequence. By methodically analyzing each function, we can pinpoint the exact representations that capture the essence of the sequence 8, 1.5, -5, -11.5...

Evaluating the Provided Functions

Now, let's meticulously evaluate each of the provided functions to determine which ones accurately represent the arithmetic sequence 8, 1.5, -5, -11.5… This involves a careful comparison of the function's structure with the sequence's characteristics, specifically the common difference and the first term.

Explicit Formulas Analysis

We'll begin with the explicit formulas, which directly link the term number (n) to the term value (f(n)). Remember, the general form for an explicit formula of an arithmetic sequence is f(n) = a + (n - 1)d, where a is the first term and d is the common difference. For our sequence, this translates to f(n) = 8 + (n - 1)(-6.5). Simplifying this, we get f(n) = 8 - 6.5n + 6.5, which further simplifies to f(n) = -6.5n + 14.5. This simplified form will be our benchmark for evaluating the given explicit functions.

1. f(n) = -6.5n + 14.5: This function matches our derived explicit formula perfectly. The coefficient of n is -6.5, which is the common difference, and the constant term is 14.5. This constant term is derived from 8 + 6.5, confirming its alignment with the sequence's initial term and common difference. Therefore, this function correctly represents the sequence.
2. f(n) = -1.5n + 9.5: In this function, the coefficient of n is -1.5, which does not match the common difference of -6.5. This immediately indicates that this function does not accurately represent the sequence. To further confirm, we can calculate the first few terms using this function: f(1) = -1.5(1) + 9.5 = 8, which matches the first term, but f(2) = -1.5(2) + 9.5 = 6.5, which does not match the second term (1.5) of the sequence. Thus, this function is incorrect.
3. f(n) = 6.5n + 1.5: This function has a positive coefficient for n (6.5), indicating an increasing sequence, whereas our sequence is decreasing. The common difference is -6.5, so this function is clearly incorrect. Calculating the first term, f(1) = 6.5(1) + 1.5 = 8, might mislead one into thinking it's correct initially, but the subsequent terms will deviate significantly. For example, f(2) = 6.5(2) + 1.5 = 14.5, which is far from the second term of the sequence (1.5).

Recursive Definitions Analysis

Next, we'll examine the recursive definitions. A recursive definition specifies the first term and a rule for finding the subsequent terms. The general form for a recursive definition of an arithmetic sequence is f(1) = a, f(n+1) = f(n) + d, where a is the first term and d is the common difference. For our sequence, this translates to f(1) = 8 and f(n+1) = f(n) - 6.5.

1. f(1) = 8, f(n+1) = f(n) - 6.5: This recursive definition perfectly matches our derived recursive form. It correctly states that the first term is 8 and that each subsequent term is obtained by subtracting 6.5 from the previous term. This aligns precisely with the arithmetic sequence 8, 1.5, -5, -11.5...
2. f(1) = 8, f(n+1) = f(n) - 1.5: This recursive definition correctly identifies the first term as 8 but uses an incorrect common difference (-1.5 instead of -6.5). This means that while the sequence will start correctly, the subsequent terms will not match the given sequence. For example, the second term would be f(2) = f(1) - 1.5 = 8 - 1.5 = 6.5, which is not the second term of our sequence (1.5).
3. f(1) = 8, f(n+1) = f(n) + 6.5: This recursive definition correctly identifies the first term as 8 but uses the opposite sign for the common difference (+6.5 instead of -6.5). This will result in an increasing sequence, whereas our sequence is decreasing. Therefore, this function does not represent the given arithmetic sequence.

Identifying the Correct Functions

Based on our meticulous analysis of both explicit formulas and recursive definitions, we can now definitively identify the functions that accurately represent the arithmetic sequence 8, 1.5, -5, -11.5…

From the explicit formulas, the function f(n) = -6.5n + 14.5 is the only one that correctly represents the sequence. This function aligns perfectly with the general form of an explicit formula for an arithmetic sequence, incorporating the common difference of -6.5 and a constant term that reflects the initial term of 8. We derived this formula by substituting the first term and the common difference into the general explicit formula and simplifying, confirming its accuracy.

Among the recursive definitions, f(1) = 8, f(n+1) = f(n) - 6.5 is the sole correct representation. This definition accurately captures the essence of the arithmetic sequence by specifying the first term as 8 and the rule for generating subsequent terms as subtracting 6.5 from the preceding term. This recursive definition mirrors the fundamental characteristic of the sequence, where each term is obtained by subtracting the common difference from the previous term.

The other functions provided, both explicit and recursive, failed to accurately represent the sequence for various reasons. Some used an incorrect common difference, while others used the wrong sign for the common difference, leading to an increasing sequence instead of a decreasing one. It's crucial to understand that the common difference is the backbone of an arithmetic sequence, and any deviation from this value will result in an incorrect representation. Similarly, the initial term serves as the anchor point, and any discrepancy here will also lead to an inaccurate depiction of the sequence.

Conclusion: Mastering Arithmetic Sequence Representations

In conclusion, accurately representing arithmetic sequences requires a thorough understanding of their underlying properties, including the common difference and the first term. The arithmetic sequence 8, 1.5, -5, -11.5… serves as an excellent example to illustrate the process of identifying the correct functions, whether explicit or recursive, that describe a given sequence.

We determined that the explicit formula f(n) = -6.5n + 14.5 and the recursive definition f(1) = 8, f(n+1) = f(n) - 6.5 are the only functions among those provided that correctly represent the sequence. This determination was made through a careful analysis of each function, comparing their structure and parameters with the sequence's characteristics. The explicit formula was derived by substituting the first term and the common difference into the general form, while the recursive definition was assessed by ensuring both the initial condition and the recursive step accurately reflected the sequence's behavior.

Mastering the representation of arithmetic sequences is a fundamental skill in mathematics. It not only enhances one's understanding of sequences and series but also lays the groundwork for more advanced concepts in algebra and calculus. The ability to translate a sequence into its mathematical representation, whether in explicit or recursive form, allows for predicting future terms, analyzing patterns, and solving real-world problems that can be modeled using arithmetic sequences. This understanding reinforces the power of mathematical notation in capturing and expressing numerical patterns, making it an invaluable tool for problem-solving and mathematical reasoning.

By dissecting the given sequence and evaluating the provided functions, we've reinforced the key principles behind arithmetic sequences and their representations. This exercise underscores the importance of precision and attention to detail when working with mathematical formulas and definitions. The correct identification of functions that represent an arithmetic sequence is a testament to a strong grasp of mathematical concepts and their applications.