Identifying The Recursive Formula For An Arithmetic Sequence

Let's dive deep into the world of arithmetic sequences and recursive formulas. This article aims to clarify how to identify the recursive formula that defines a given arithmetic sequence. We'll break down the key concepts and provide a step-by-step approach to solving this type of problem.

The problem presents us with three terms of an arithmetic sequence: $f(1) = 6$, $f(4) = 12$, and $f(7) = 18$. Our goal is to determine which of the provided recursive formulas accurately describes this sequence. The options are:

A. $f(n+1) = f(n) + 6$ B. $f(n+1) = 2f(n)$ C. $f(n+1) = f(n) + 2$ D. $f(n+1) = 1.5f(n)$

Decoding Arithmetic Sequences

Before we tackle the problem, it’s crucial to understand what an arithmetic sequence is. An arithmetic sequence is a series of numbers where the difference between consecutive terms remains constant. This constant difference is known as the common difference. For instance, the sequence 2, 4, 6, 8, 10... is an arithmetic sequence with a common difference of 2.

To find the common difference in a given arithmetic sequence, you simply subtract any term from the term that follows it. In our case, we have three terms: $f(1) = 6$, $f(4) = 12$, and $f(7) = 18$. Notice that the terms are not consecutive, but we can still deduce the common difference.

First, let's find the difference between the positions of the terms. The difference between the positions of $f(1)$ and $f(4)$ is $4 - 1 = 3$. The difference in their values is $12 - 6 = 6$. Therefore, over three terms, the sequence increases by 6. This means the common difference per term is $6 / 3 = 2$.

Similarly, the difference between the positions of $f(4)$ and $f(7)$ is $7 - 4 = 3$. The difference in their values is $18 - 12 = 6$. Again, over three terms, the sequence increases by 6, confirming that the common difference is indeed 2.

Understanding the common difference is paramount to identifying the correct recursive formula. This difference dictates how each term relates to the previous one.

Understanding Recursive Formulas

A recursive formula defines a sequence by specifying how each term is related to the preceding term(s). It typically consists of two parts:

1. The initial term(s): This provides the starting point of the sequence. In our problem, we have $f(1) = 6$, which serves as our initial term.
2. The recursive step: This is the rule that defines how to calculate the next term based on the previous term(s). This is usually expressed in the form $f(n+1) = ...$, where $f(n+1)$ represents the next term, and $f(n)$ represents the current term.

The recursive formulas provided in the options illustrate this concept. Each formula tells us how to find the next term, $f(n+1)$, based on the current term, $f(n)$. The key is to identify the formula that accurately reflects the common difference we calculated earlier.

For example, the formula $f(n+1) = f(n) + 6$ suggests that each term is obtained by adding 6 to the previous term. The formula $f(n+1) = 2f(n)$ indicates that each term is twice the previous term. The correct formula will align with the arithmetic sequence we are given.

Analyzing the Options

Now, let's examine each option in light of our understanding of arithmetic sequences and recursive formulas:

• A. $f(n+1) = f(n) + 6$

  This formula implies that we add 6 to the previous term to get the next term. Let’s test this with our sequence. We know $f(1) = 6$. According to this formula, $f(2) = f(1) + 6 = 6 + 6 = 12$. However, we don't have $f(2)$ in our given sequence, but we do have $f(4) = 12$. This formula doesn't immediately seem correct because the common difference is not 6 between each consecutive term. This adds 6 over three terms ($f(1)$ to $f(4)$), not one term.

• B. $f(n+1) = 2f(n)$

  This formula suggests that each term is twice the previous term. If we start with $f(1) = 6$, then $f(2) = 2 * 6 = 12$. This seems to match $f(4) = 12$, but let’s check further. $f(3)$ would be $2 * f(2) = 2 * 12 = 24$, and $f(4)$ would be $2 * f(3) = 2 * 24 = 48$. This clearly doesn't match our given $f(4) = 12$. So, this formula is incorrect. This formula represents a geometric sequence, not an arithmetic sequence.

• C. $f(n+1) = f(n) + 2$

  This formula indicates that we add 2 to the previous term to get the next term. This aligns perfectly with our calculated common difference of 2. Let’s test it. Starting with $f(1) = 6$, we would have:

  • $f(2) = f(1) + 2 = 6 + 2 = 8$
  • $f(3) = f(2) + 2 = 8 + 2 = 10$
  • $f(4) = f(3) + 2 = 10 + 2 = 12$ (This matches our given $f(4)$)
  • $f(5) = f(4) + 2 = 12 + 2 = 14$
  • $f(6) = f(5) + 2 = 14 + 2 = 16$
  • $f(7) = f(6) + 2 = 16 + 2 = 18$ (This matches our given $f(7)$)

  This formula holds true for the given terms in the sequence.

• D. $f(n+1) = 1.5f(n)$

  This formula suggests that each term is 1.5 times the previous term. Starting with $f(1) = 6$, we get $f(2) = 1.5 * 6 = 9$. Then, $f(3) = 1.5 * 9 = 13.5$, and $f(4) = 1.5 * 13.5 = 20.25$. This doesn't match our given $f(4) = 12$, so this formula is incorrect. This is another example of a geometric sequence, not an arithmetic sequence.

The Definitive Answer

Based on our analysis, the recursive formula that accurately defines the given arithmetic sequence is C. $f(n+1) = f(n) + 2$. This formula correctly captures the constant difference of 2 between consecutive terms.

Key Takeaways for Recursive Formulas

• Identify the common difference: In an arithmetic sequence, the common difference is the constant value added to each term to get the next term. This is critical for determining the correct recursive formula.
• Understand the structure of a recursive formula: A recursive formula defines a term based on the preceding term(s). It typically includes an initial term and a recursive step.
• Test the formula: Once you've identified a potential recursive formula, test it with the given terms of the sequence to ensure it holds true.
• Distinguish arithmetic from geometric: Be mindful of whether the sequence is arithmetic (constant difference) or geometric (constant ratio). The recursive formulas will differ significantly.

By following these steps and understanding the fundamental concepts, you can confidently tackle problems involving arithmetic sequences and recursive formulas. Remember that practice is key to mastering these concepts, so work through various examples to solidify your understanding.

This article explored the essential components of arithmetic sequences and recursive formulas, providing a detailed solution to the presented problem. We hope this has enhanced your understanding and equipped you with the skills to approach similar challenges in the future. Remember, mathematics is a journey of discovery, and each problem solved is a step forward in your learning process.