Explicit Formula Equivalent To A₁=7, Aₙ=aₙ₋₁+3.2

This article delves into the fascinating world of arithmetic sequences, specifically addressing the question: Which explicit formula is equivalent to the recursive definition a₁ = 7, aₙ = aₙ₋₁ + 3.2? We'll break down the components of arithmetic sequences, explore how to derive explicit formulas from recursive definitions, and ultimately identify the correct formula from the given options. Understanding the nuances of arithmetic sequences is crucial not only for mathematical problem-solving but also for appreciating patterns that emerge in various real-world scenarios.

Understanding Arithmetic Sequences

At the heart of this problem lies the concept of an arithmetic sequence. An arithmetic sequence is a series of numbers where the difference between any two consecutive terms remains constant. This constant difference is known as the common difference, often denoted by d. The sequence starts with an initial term, a₁, and each subsequent term is generated by adding the common difference to the preceding term. In the given recursive definition, a₁ = 7 tells us that the first term of the sequence is 7. The equation aₙ = aₙ₋₁ + 3.2 defines the relationship between consecutive terms; it states that to find any term (aₙ), you simply add 3.2 (the common difference) to the previous term (aₙ₋₁). This type of definition, which relies on the preceding term to define the current term, is called a recursive definition.

The beauty of arithmetic sequences lies in their predictable nature. This predictability allows us to define them in two primary ways: recursively and explicitly. As we've seen, the recursive definition provides a step-by-step method for generating the sequence. However, it's often more convenient to have an explicit formula, which directly calculates the nth term (aₙ) based on its position (n) in the sequence. This explicit formula eliminates the need to know the previous terms, making it incredibly useful for finding terms far down the sequence. For example, if we wanted to find the 100th term of the sequence using the recursive definition, we would need to calculate the first 99 terms first. However, with an explicit formula, we can directly plug in n = 100 and obtain the result. Therefore, understanding how to convert a recursive definition into an explicit formula is a fundamental skill in working with arithmetic sequences. The explicit formula not only simplifies calculations but also provides deeper insights into the sequence's overall behavior and structure. It allows us to visualize the sequence as a linear function, connecting the world of sequences to the broader realm of linear algebra and calculus.

Deriving the Explicit Formula

The core task before us is to find the explicit formula equivalent to the given recursive definition. To accomplish this, let's explore the general form of an explicit formula for an arithmetic sequence. The general explicit formula is given by: aₙ = a₁ + (n - 1)d, where aₙ represents the nth term, a₁ is the first term, n is the term number, and d is the common difference. This formula elegantly captures the essence of an arithmetic sequence: each term is simply the first term plus a multiple of the common difference. The (n - 1) factor arises because we start counting the common difference from the second term onwards. In other words, the first term (a₁) has zero common differences added to it, the second term (a₂) has one common difference added, the third term (a₃) has two common differences added, and so on.

Now, let's apply this general formula to our specific case. We are given that a₁ = 7 (the first term) and the common difference d = 3.2. Substituting these values into the general formula, we get: aₙ = 7 + (n - 1)3.2. This formula provides a direct way to calculate any term in the sequence. For instance, to find the 5th term (a₅), we would simply substitute n = 5 into the formula: a₅ = 7 + (5 - 1)3.2 = 7 + 4 * 3.2 = 7 + 12.8 = 19.8. The explicit formula not only offers computational convenience but also reveals the linear relationship inherent in arithmetic sequences. If we were to plot the terms of the sequence on a graph, with n on the x-axis and aₙ on the y-axis, we would obtain a straight line with a slope equal to the common difference (d) and a y-intercept related to the first term (a₁). This connection between arithmetic sequences and linear functions provides a powerful visual and analytical tool for understanding their behavior and properties. Furthermore, the explicit formula serves as a bridge to more advanced mathematical concepts, such as series and summations, where we can use it to derive formulas for the sum of the first n terms of an arithmetic sequence.

Evaluating the Given Options

With the derived explicit formula in hand, we can now evaluate the given options to identify the correct one. Let's revisit the provided options:

• aₙ = 7(3.2)ⁿ⁻¹
• aₙ = aₙ₋₁ + (n - 1)3.2
• aₙ = 7 + (n - 1)3.2
• aₙ = 3.2 + (n - 1)7

Option 1, aₙ = 7(3.2)ⁿ⁻¹, represents a geometric sequence, not an arithmetic sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, whereas in an arithmetic sequence, we add a constant difference. Therefore, this option is incorrect.

Option 2, aₙ = aₙ₋₁ + (n - 1)3.2, is a recursive formula, similar in nature to the original given recursive definition. However, it's not the correct explicit formula. This formula still relies on the previous term (aₙ₋₁) to calculate the current term (aₙ), which is a characteristic of recursive formulas, not explicit formulas.

Option 3, aₙ = 7 + (n - 1)3.2, perfectly matches the explicit formula we derived using the general form aₙ = a₁ + (n - 1)d. This formula correctly incorporates the first term (a₁ = 7) and the common difference (d = 3.2). Therefore, this is the correct answer.

Option 4, aₙ = 3.2 + (n - 1)7, while having a similar structure to the explicit formula, incorrectly swaps the roles of the first term and the common difference. This formula would represent an arithmetic sequence with a first term of 3.2 and a common difference of 7, which is not the sequence defined by the given recursive definition. By carefully comparing each option with the derived explicit formula and understanding the characteristics of arithmetic sequences, we can confidently identify the correct answer. This process reinforces the importance of not only knowing the formulas but also comprehending their underlying principles and how they relate to the properties of the sequences they represent.

Conclusion

In conclusion, the explicit formula equivalent to the recursive definition a₁ = 7, aₙ = aₙ₋₁ + 3.2 is aₙ = 7 + (n - 1)3.2. This solution was reached by understanding the definition of arithmetic sequences, deriving the general explicit formula, and substituting the given values. The ability to convert between recursive and explicit formulas is a crucial skill in mathematics, enabling us to analyze and predict the behavior of sequences and patterns. Furthermore, this exercise highlights the interconnectedness of mathematical concepts, linking arithmetic sequences to linear functions and showcasing the power of explicit formulas in simplifying calculations and revealing underlying structures. By mastering these concepts, we can confidently tackle more complex mathematical problems and appreciate the elegance and predictability inherent in the world of sequences and patterns.