Explicit Function For Arithmetic Sequence Explained
#1. Introduction
In mathematics, sequences play a crucial role in understanding patterns and relationships between numbers. Among these, arithmetic sequences are particularly important due to their consistent and predictable nature. 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 and expressing arithmetic sequences using explicit functions is a fundamental skill in algebra and calculus. This article will guide you through the process of finding the explicit function for a given arithmetic sequence, using a specific example to illustrate the steps involved. Understanding explicit functions allows us to determine any term in the sequence without having to list all the preceding terms, making it a powerful tool for mathematical analysis and problem-solving. To fully grasp this concept, it’s essential to first understand the basic terminology and formulas associated with arithmetic sequences. The explicit formula, also known as the general formula, is the key to expressing any term in the sequence directly in terms of its position. This formula is derived from the fundamental properties of arithmetic sequences and provides a concise way to represent the sequence. In the following sections, we will explore the core concepts of arithmetic sequences, the derivation of the explicit formula, and how to apply it to solve problems, ensuring a comprehensive understanding of this topic.
#2. Understanding Arithmetic Sequences
To effectively tackle the problem of finding the explicit function for an arithmetic sequence, it's crucial to first understand what an arithmetic sequence is and its key characteristics. An arithmetic sequence is a series of numbers in which the difference between consecutive terms remains constant. This constant difference is called the common difference, often denoted as d. For example, in the sequence 2, 5, 8, 11, 14, the common difference is 3 because each term is obtained by adding 3 to the previous term. The first term of an arithmetic sequence is typically denoted as a₁, and the nth term is denoted as aₙ. The general form of an arithmetic sequence can be written as: a₁, a₁ + d, a₁ + 2d, a₁ + 3d, ..., a₁ + (n-1)d, ... Understanding the common difference is essential because it is the backbone of an arithmetic sequence, dictating how the sequence progresses. Identifying the common difference is often the first step in analyzing and working with arithmetic sequences. The formula for the nth term of an arithmetic sequence is a direct result of repeatedly adding the common difference to the first term. This formula is the basis for deriving the explicit function, which we will discuss in detail later. Being able to recognize and work with arithmetic sequences is a fundamental skill in mathematics, with applications ranging from simple number patterns to more complex topics like series and calculus. The ability to identify the common difference, the first term, and any other term in the sequence is crucial for understanding the behavior and properties of arithmetic sequences. In the next section, we will delve into the explicit formula for arithmetic sequences, which is the key to expressing any term in the sequence directly in terms of its position.
#3. The Explicit Formula for Arithmetic Sequences
The explicit formula is a powerful tool for working with arithmetic sequences. It allows us to find any term in the sequence directly, without needing to know the previous terms. The explicit formula for the nth term (aₙ) of an arithmetic sequence is given by: aₙ = a₁ + (n - 1)d where: aₙ is the nth term of the sequence, a₁ is the first term of the sequence, n is the position of the term in the sequence (e.g., 1 for the first term, 2 for the second term, and so on), and d is the common difference between consecutive terms. This formula is derived from the basic definition of an arithmetic sequence, where each term is obtained by adding the common difference to the previous term. The (n - 1) factor in the formula represents the number of times the common difference is added to the first term to reach the nth term. To illustrate, let’s consider a simple arithmetic sequence: 3, 7, 11, 15, ... Here, the first term (a₁) is 3, and the common difference (d) is 4. Using the explicit formula, we can find the 10th term (a₁₀) as follows: a₁₀ = 3 + (10 - 1) * 4 = 3 + 9 * 4 = 3 + 36 = 39. This demonstrates how the explicit formula allows us to quickly calculate any term in the sequence. The explicit formula is particularly useful when dealing with large values of n, where calculating each term individually would be impractical. It provides a direct relationship between the term number and the term value, making it a fundamental tool for analyzing and working with arithmetic sequences. Understanding and applying the explicit formula is crucial for solving various problems related to arithmetic sequences, including finding specific terms, determining if a given number is part of the sequence, and even constructing the sequence based on given information. In the next section, we will apply this formula to the specific problem at hand, finding the explicit function for the given arithmetic sequence.
#4. Applying the Explicit Formula to the Given Sequence
Now, let's apply the explicit formula to the given arithmetic sequence: -351, -343, -335, -327, -319. The first step is to identify the first term (a₁) and the common difference (d). From the sequence, we can see that the first term (a₁) is -351. To find the common difference (d), we subtract any term from its subsequent term. For example: d = -343 - (-351) = -343 + 351 = 8. So, the common difference (d) is 8. Now that we have a₁ and d, we can plug these values into the explicit formula: aₙ = a₁ + (n - 1)d aₙ = -351 + (n - 1) * 8. Next, we simplify the expression: aₙ = -351 + 8n - 8 aₙ = 8n - 359. This is the explicit formula for the given arithmetic sequence. It expresses the nth term (aₙ) as a function of n. To verify this formula, we can plug in values for n and check if the resulting terms match the given sequence. For example, for n = 1: a₁ = 8(1) - 359 = 8 - 359 = -351, which is the first term. For n = 2: a₂ = 8(2) - 359 = 16 - 359 = -343, which is the second term. This confirms that our explicit formula is correct. The explicit formula, aₙ = 8n - 359, allows us to find any term in the sequence directly. For instance, to find the 10th term, we simply plug in n = 10: a₁₀ = 8(10) - 359 = 80 - 359 = -279. This demonstrates the power of the explicit formula in efficiently determining terms in an arithmetic sequence. In the next section, we will discuss how to express this explicit formula as a function, which is the final step in solving the original problem.
#5. Expressing the Explicit Formula as a Function
In the previous section, we derived the explicit formula for the given arithmetic sequence as aₙ = 8n - 359. To express this formula as a function, we simply replace aₙ with f(n). Therefore, the explicit function that defines the arithmetic sequence is: f(n) = 8n - 359 This function represents the same relationship as the explicit formula, but it is written in function notation, which is commonly used in mathematics. The function f(n) takes the term number n as input and returns the corresponding term in the sequence as output. The explicit function provides a concise and clear way to represent the arithmetic sequence. It allows us to easily calculate any term in the sequence by plugging in the appropriate value for n. For example, to find the 5th term, we simply evaluate f(5): f(5) = 8(5) - 359 = 40 - 359 = -319, which matches the 5th term in the given sequence. The explicit function is a powerful tool for analyzing and working with arithmetic sequences. It encapsulates the pattern of the sequence in a single equation, making it easy to understand and manipulate. In the context of the original problem, the question asks for the explicit function that defines the arithmetic sequence. We have successfully derived this function by first finding the common difference and the first term, then applying the explicit formula, and finally expressing it in function notation. The function f(n) = 8n - 359 is the solution to the problem. This function allows us to find any term in the sequence, making it a complete and effective representation of the arithmetic sequence. In the final section, we will review the entire process and highlight the key steps involved in finding the explicit function for an arithmetic sequence.
#6. Conclusion and Review
In this article, we have explored the process of finding the explicit function for an arithmetic sequence. We started by understanding the basic concepts of arithmetic sequences, including the definition of a common difference and the general form of an arithmetic sequence. We then delved into the explicit formula, aₙ = a₁ + (n - 1)d, which allows us to find any term in the sequence directly. Applying this formula to the given sequence, -351, -343, -335, -327, -319, we identified the first term (a₁) as -351 and the common difference (d) as 8. Plugging these values into the explicit formula, we obtained aₙ = -351 + (n - 1) * 8. Simplifying this expression, we arrived at aₙ = 8n - 359. Finally, we expressed this explicit formula as a function, f(n) = 8n - 359, which is the solution to the problem. This function defines the arithmetic sequence and allows us to find any term in the sequence by plugging in the appropriate value for n. The key steps involved in finding the explicit function for an arithmetic sequence are: Identify the first term (a₁) of the sequence. Calculate the common difference (d) by subtracting any term from its subsequent term. Apply the explicit formula: aₙ = a₁ + (n - 1)d. Simplify the expression to obtain the explicit formula. Express the explicit formula as a function by replacing aₙ with f(n). Understanding and applying these steps is crucial for working with arithmetic sequences and solving related problems. The explicit function is a powerful tool for representing and analyzing arithmetic sequences, providing a concise and effective way to understand their patterns and properties. By mastering this concept, you can confidently tackle a wide range of mathematical problems involving arithmetic sequences.
Therefore, the correct answer is D. f(n) = 8n - 359
