Finding The Explicit Formula For The Sequence -7, -4, -1, 2, 5

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Finding the explicit formula for a sequence is a fundamental concept in mathematics, particularly when dealing with arithmetic sequences. In this article, we will delve into the process of determining the explicit formula for the sequence -7, -4, -1, 2, 5. This involves identifying the pattern within the sequence, calculating the common difference, and constructing a formula that allows us to find any term in the sequence directly. We will explore the various options provided and methodically eliminate the incorrect choices to arrive at the correct explicit formula. Understanding how to derive explicit formulas is crucial for solving various mathematical problems, making predictions about future terms, and analyzing the behavior of sequences in general. By the end of this article, you will have a comprehensive understanding of how to approach such problems and confidently determine the explicit formula for arithmetic sequences.

Understanding Arithmetic Sequences

To determine the explicit formula for the sequence -7, -4, -1, 2, 5, it’s essential to first understand what an arithmetic sequence is. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference. For example, the sequence 2, 4, 6, 8, 10 is an arithmetic sequence because the common difference between each term is 2. Recognizing whether a sequence is arithmetic is the first step in finding its explicit formula. In our case, the sequence -7, -4, -1, 2, 5 appears to be arithmetic, but we need to confirm this by calculating the differences between consecutive terms.

Identifying the Common Difference

The common difference is the key to unlocking the explicit formula for an arithmetic sequence. To find the common difference (often denoted as 'd'), we subtract any term from the term that follows it. Let's apply this to our sequence: -7, -4, -1, 2, 5.

  • The difference between -4 and -7 is -4 - (-7) = 3.
  • The difference between -1 and -4 is -1 - (-4) = 3.
  • The difference between 2 and -1 is 2 - (-1) = 3.
  • The difference between 5 and 2 is 5 - 2 = 3.

Since the difference between each pair of consecutive terms is consistently 3, we can confirm that the sequence is arithmetic, and the common difference, d, is 3. This consistent difference is crucial because it forms the basis of our explicit formula. Without a common difference, the sequence would not be arithmetic, and a different approach would be needed to find its formula. Understanding this common difference allows us to predict how the sequence will continue and is a fundamental component in constructing the explicit formula.

The General Form of an Explicit Formula

Knowing that our sequence is arithmetic with a common difference of 3, we can now consider the general form of an explicit formula for an arithmetic sequence. The explicit formula, also known as the nth term formula, allows us to find any term in the sequence directly without having to list out all the preceding terms. The general form of the explicit formula for an arithmetic sequence is:

a_n = a_1 + (n - 1)d

Where:

  • a_n is the nth term in the sequence (the term we want to find).
  • a_1 is the first term in the sequence.
  • n is the term number (the position of the term in the sequence).
  • d is the common difference.

This formula essentially states that any term in the sequence is equal to the first term plus the common difference multiplied by one less than the term number. This makes intuitive sense because each term is built upon the previous one by adding the common difference. Understanding this general form is vital because it provides a template into which we can plug in our specific values to find the explicit formula for our sequence. In our sequence, a_1 is -7 and d is 3, which we will use to construct our specific formula.

Applying the General Form to Our Sequence

Now that we have the general form of the explicit formula, a_n = a_1 + (n - 1)d, we can substitute the values from our sequence -7, -4, -1, 2, 5 to find the specific formula. We know that the first term, a_1, is -7, and the common difference, d, is 3. Plugging these values into the general formula, we get:

a_n = -7 + (n - 1)3

This equation now represents the explicit formula for our specific sequence. It tells us exactly how to find any term in the sequence by knowing its position (n). For instance, if we wanted to find the 10th term, we would substitute n = 10 into the formula. This formula is a powerful tool because it encapsulates the entire sequence in a concise equation. It's essential to understand this step because it bridges the gap between the general theory of arithmetic sequences and the practical application of finding formulas for specific sequences. Next, we will compare our derived formula with the options provided to determine the correct answer.

Evaluating the Given Options

We have derived the explicit formula a_n = -7 + (n - 1)3 for the sequence -7, -4, -1, 2, 5. Now, we need to compare this formula with the options provided to identify the correct answer. The options given are:

  • A. a_n = -7 + (n - 1)(-3)
  • B. a_n = 8 + (n - 1)3
  • C. a_n = -7 + (n - 1)3
  • D. a_n = 3 + (n - 1)(-7)

By comparing our derived formula with the options, we can see that option C, a_n = -7 + (n - 1)3, exactly matches our result. Options A, B, and D have different structures or values that do not align with our calculations. For example, option A has a common difference of -3, which is incorrect for our sequence. Option B has a different initial term (8 instead of -7) and is therefore also incorrect. Option D has both an incorrect initial term (3) and an incorrect common difference (-7). This step of comparing and contrasting is crucial because it ensures that we not only understand how to derive the formula but also how to recognize it among other similar expressions. The correct identification of the formula confirms our understanding of the sequence and the explicit formula.

Why Other Options Are Incorrect

To solidify our understanding, let’s examine why the other options are incorrect. This involves analyzing each option and demonstrating how it fails to accurately represent the sequence -7, -4, -1, 2, 5.

  • Option A: a_n = -7 + (n - 1)(-3)
    • This formula suggests a common difference of -3. If we apply this formula to find the second term (n = 2), we get a_2 = -7 + (2 - 1)(-3) = -7 - 3 = -10. However, the second term in our sequence is -4, not -10. Therefore, this formula does not match the sequence.
  • Option B: a_n = 8 + (n - 1)3
    • This formula suggests a first term of 8. In our sequence, the first term is -7. Even if the common difference of 3 were correct, the starting point is incorrect. If we calculate the first term (n = 1) using this formula, we get a_1 = 8 + (1 - 1)3 = 8, which is clearly not the first term of our sequence.
  • Option D: a_n = 3 + (n - 1)(-7)
    • This formula has both an incorrect first term (3) and an incorrect common difference (-7). Applying this formula to find the second term (n = 2), we get a_2 = 3 + (2 - 1)(-7) = 3 - 7 = -4. While the second term happens to match, this is coincidental. If we calculate the third term (n = 3), we get a_3 = 3 + (3 - 1)(-7) = 3 - 14 = -11, which does not match the third term in our sequence (-1). Therefore, this formula is incorrect.

By demonstrating why these options fail, we reinforce our understanding of how the explicit formula should accurately reflect both the initial term and the common difference of the sequence. This detailed analysis solidifies our confidence in the correct answer.

Conclusion: The Correct Explicit Formula

In conclusion, the correct explicit formula for the sequence -7, -4, -1, 2, 5 is C. a_n = -7 + (n - 1)3. We arrived at this answer by first identifying the sequence as arithmetic, calculating the common difference to be 3, and recognizing the first term as -7. We then applied the general form of the explicit formula for arithmetic sequences, a_n = a_1 + (n - 1)d, substituting the values to get a_n = -7 + (n - 1)3. Finally, we compared our derived formula with the given options and confirmed that option C was the only accurate representation of the sequence.

Understanding how to find explicit formulas is a crucial skill in mathematics, enabling us to predict and analyze patterns within sequences. This process involves recognizing arithmetic sequences, determining the common difference, applying the general formula, and validating the result. By mastering these steps, you can confidently tackle similar problems and gain a deeper understanding of sequences and their formulas.