Arithmetic sequences are fundamental concepts in mathematics, particularly in the study of sequences and series. Understanding how to identify and extend these sequences is a crucial skill for anyone delving into mathematical problem-solving. In this article, we will explore the arithmetic sequence -18, -13, -8, -3 and determine the next three terms. This involves identifying the common difference between consecutive terms and applying it to find subsequent terms. We'll break down the process step by step, ensuring a clear understanding of how to tackle similar problems. By the end of this article, you'll be well-equipped to handle arithmetic sequences and confidently find missing terms.
Understanding Arithmetic Sequences
To fully grasp the task at hand, let's first define 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 known as the common difference. Recognizing an arithmetic sequence is the first step in solving problems that involve finding the next terms. The general form of an arithmetic sequence can be expressed as:
a, a + d, a + 2d, a + 3d, ...
where:
ais the first term,dis the common difference.
To identify if a sequence is arithmetic, you simply subtract any term from its subsequent term. If the result is the same throughout the sequence, then it is an arithmetic sequence. For example, in the sequence 2, 5, 8, 11, the difference between consecutive terms is consistently 3 (5-2 = 3, 8-5 = 3, 11-8 = 3). Therefore, this is an arithmetic sequence with a common difference of 3. Understanding this foundational concept is vital before we tackle the given sequence.
Identifying the Common Difference
In the given sequence -18, -13, -8, -3, our initial task is to confirm that it is indeed an arithmetic sequence and, if so, to find the common difference. We do this by subtracting each term from its subsequent term:
- -13 - (-18) = -13 + 18 = 5
- -8 - (-13) = -8 + 13 = 5
- -3 - (-8) = -3 + 8 = 5
As we can see, the difference between each pair of consecutive terms is consistently 5. This confirms that the given sequence is an arithmetic sequence, and the common difference (d) is 5. This is a crucial piece of information, as the common difference is the key to finding the subsequent terms in the sequence. Now that we have established the common difference, we can confidently move forward to calculating the next three terms. This methodical approach ensures that we understand the underlying principles and apply them correctly.
Finding the Next Three Terms
Now that we've established that the sequence -18, -13, -8, -3 is arithmetic with a common difference of 5, we can proceed to find the next three terms. The process is straightforward: we simply add the common difference to the last known term to get the next term, and repeat this process for the subsequent terms. This iterative method allows us to extend the sequence as far as needed. Let's start by finding the next term after -3.
Step-by-Step Calculation
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First Term: To find the term immediately following -3, we add the common difference (5) to -3:
-3 + 5 = 2So, the next term in the sequence is 2.
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Second Term: Now, we add the common difference (5) to the term we just found (2):
2 + 5 = 7Thus, the term after 2 is 7.
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Third Term: Finally, we add the common difference (5) to the term 7:
7 + 5 = 12Therefore, the term following 7 is 12.
By following this systematic approach, we have successfully found the next three terms of the arithmetic sequence. These terms are 2, 7, and 12. This simple yet effective method can be applied to any arithmetic sequence, making it a valuable tool in mathematical problem-solving. Let's now consolidate our findings and present the complete sequence with the newly found terms.
The Extended Sequence
Having calculated the next three terms, we can now extend the original sequence -18, -13, -8, -3. The extended sequence, including the terms we just found, is:
-18, -13, -8, -3, 2, 7, 12
This expanded sequence clearly demonstrates the arithmetic progression, with each term increasing by 5. This exercise not only provides the answer to the problem but also reinforces the concept of arithmetic sequences and the predictability they offer. Recognizing patterns and applying simple arithmetic operations can unlock solutions to a variety of mathematical problems. Now, let's consider the given options and identify which one correctly presents the next three terms of the sequence.
Evaluating the Options
Now that we have determined the next three terms of the arithmetic sequence to be 2, 7, and 12, we need to compare our answer with the provided options. This step is crucial to ensure that we select the correct answer and avoid any potential errors. Let's examine the options given:
A. -8, -13, -18 B. 3, 8, 13 C. 2, 7, 12 D. 0, 5, 10
By comparing our calculated terms (2, 7, 12) with the options, it becomes clear that option C perfectly matches our result. Options A, B, and D present different sequences that do not align with the arithmetic progression we've established. This comparative analysis underscores the importance of not only finding the solution but also verifying it against the given choices. Let's formally state our final answer, emphasizing the correct option and the corresponding terms.
The Correct Answer
After careful calculation and comparison, the correct answer is:
C. 2, 7, 12
This option accurately represents the next three terms of the arithmetic sequence -18, -13, -8, -3. Our step-by-step approach, from identifying the common difference to calculating subsequent terms, has led us to this definitive solution. This exercise highlights the practical application of arithmetic sequences and demonstrates how a methodical approach can lead to accurate results. In the next section, we will summarize the entire process, reinforcing the key concepts and steps involved in solving such problems.
Summary and Conclusion
In this article, we addressed the problem of finding the next three terms in the arithmetic sequence -18, -13, -8, -3. We began by defining arithmetic sequences and emphasizing the importance of the common difference. Our step-by-step approach involved:
- Identifying the common difference (which was found to be 5).
- Calculating the subsequent terms by adding the common difference to the last known term.
- Finding the next three terms: 2, 7, and 12.
- Evaluating the provided options and confirming that option C (2, 7, 12) was the correct answer.
This exercise underscores the fundamental principles of arithmetic sequences and provides a clear methodology for solving similar problems. By understanding and applying these principles, you can confidently tackle arithmetic sequence problems in various mathematical contexts. Arithmetic sequences are not just abstract mathematical concepts; they appear in various real-world applications, from calculating simple interest to predicting patterns in nature. Therefore, mastering these concepts is a valuable skill for anyone interested in mathematics and its applications. Through practice and understanding, problems involving arithmetic sequences can become straightforward and even enjoyable to solve.