Finding The 18th Term In An Arithmetic Sequence A Step-by-Step Guide

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Finding a specific term in a sequence, especially the 18th term, requires understanding the underlying pattern. In this article, we will delve into the world of arithmetic sequences, providing a comprehensive guide on how to identify them and calculate any term within the sequence. Our focus will be on the given sequence: -21, -14, -7, 0, 7, ..., and we will systematically determine the 18th term. This exploration will not only enhance your understanding of arithmetic sequences but also equip you with the tools to solve similar problems.

Understanding Arithmetic Sequences

Arithmetic sequences are fundamental in mathematics, representing ordered lists of numbers where the difference between consecutive terms remains constant. This constant difference is known as the common difference. Recognizing an arithmetic sequence is the first step in finding any specific term within it. To identify an arithmetic sequence, examine the differences between successive terms. If these differences are consistent, you are dealing with an arithmetic sequence.

In our given sequence, -21, -14, -7, 0, 7, ..., let's calculate the differences:

  • -14 - (-21) = 7
  • -7 - (-14) = 7
  • 0 - (-7) = 7
  • 7 - 0 = 7

The common difference is consistently 7. This confirms that the sequence is indeed an arithmetic sequence. Understanding this foundational concept is crucial because it allows us to apply specific formulas designed for arithmetic sequences. These formulas simplify the process of finding any term, regardless of its position in the sequence. The ability to identify and work with arithmetic sequences is a key skill in algebra and various other mathematical contexts, making it an essential topic for students and enthusiasts alike. This foundational understanding paves the way for more complex sequence and series concepts.

Determining the Common Difference

The common difference is the backbone of any arithmetic sequence, and finding it is pivotal for calculating any term within the sequence. As established, the common difference is the constant value added to each term to obtain the next term. To determine this, subtract any term from its succeeding term. This simple yet crucial step unlocks the arithmetic sequence's pattern, enabling us to predict future terms.

In the sequence -21, -14, -7, 0, 7, ..., we can choose any two consecutive terms to find the common difference. For instance, let's subtract the first term (-21) from the second term (-14):

Common Difference (d) = -14 - (-21) = 7

We can verify this by subtracting any other pair of consecutive terms:

  • -7 - (-14) = 7
  • 0 - (-7) = 7
  • 7 - 0 = 7

As demonstrated, the common difference (d) is consistently 7. This consistent difference not only confirms that the sequence is arithmetic but also provides us with the value needed for further calculations. The common difference acts as a building block, allowing us to move from one term to the next within the sequence. Without accurately determining the common difference, finding a specific term like the 18th term would be significantly more challenging. The common difference is, therefore, an indispensable element in solving arithmetic sequence problems and understanding the sequence's behavior.

The Formula for the nth Term

The formula for the nth term of an arithmetic sequence is a powerful tool that allows us to calculate any term in the sequence without having to list out all the preceding terms. This formula is expressed as:

  • an = a1 + (n - 1)d

Where:

  • an represents the nth term we want to find.
  • a1 is the first term of the sequence.
  • n is the term number (e.g., for the 18th term, n = 18).
  • d is the common difference between terms.

This formula encapsulates the essence of an arithmetic sequence, illustrating how each term is derived from the first term and the common difference. It highlights that the nth term is obtained by adding the common difference (n-1) times to the first term. This is because, from the first term to the nth term, there are (n-1) steps, each adding the common difference.

To effectively use this formula, you need to identify the first term (a1), the common difference (d), and the term number (n) you are interested in. Once you have these values, simply plug them into the formula and perform the calculation. The formula is not just a mathematical expression; it's a representation of the sequence's growth pattern. By understanding and applying this formula, you can efficiently solve a wide range of problems related to arithmetic sequences, making it an essential concept in algebra and beyond. Its elegance lies in its simplicity and its ability to provide a direct path to finding any term in the sequence.

Applying the Formula to Find the 18th Term

Now, let's put the formula into action to find the 18th term of the sequence -21, -14, -7, 0, 7, .... We have already identified the key components needed for the formula:

  • First term (a1) = -21
  • Common difference (d) = 7
  • Term number (n) = 18 (since we want to find the 18th term)

Plugging these values into the formula:

  • an = a1 + (n - 1)d
  • a18 = -21 + (18 - 1) * 7

Now, let's perform the calculation step by step:

  • a18 = -21 + (17) * 7
  • a18 = -21 + 119
  • a18 = 98

Therefore, the 18th term of the arithmetic sequence is 98. This process demonstrates the power and efficiency of the formula. Instead of manually adding the common difference 17 times, we used the formula to directly calculate the 18th term. This method is especially valuable when dealing with sequences where you need to find terms far down the line, as it saves significant time and effort. Applying the formula correctly involves careful substitution and arithmetic, but once mastered, it becomes an indispensable tool for solving arithmetic sequence problems. The result, 98, not only answers the specific question but also reinforces our understanding of how arithmetic sequences progress and how we can predict their behavior.

Conclusion

In conclusion, finding the 18th term of the arithmetic sequence -21, -14, -7, 0, 7, ... involved a systematic approach that highlights the key characteristics of arithmetic sequences. We began by understanding the definition of an arithmetic sequence and how to identify the common difference, a critical step in solving such problems. We then introduced the formula for the nth term, an = a1 + (n - 1)d, which is a powerful tool for finding any term in an arithmetic sequence. By correctly identifying the first term (a1), the common difference (d), and the term number (n), we were able to apply the formula effectively.

Through the step-by-step calculation, we determined that the 18th term of the sequence is 98. This result showcases the efficiency and accuracy of the formula, especially when dealing with higher-numbered terms. The ability to find a specific term without having to list out all the preceding terms is a significant advantage, particularly in more complex problems. Understanding and applying this method not only helps in solving specific problems but also enhances your overall grasp of arithmetic sequences and their behavior. The principles and techniques discussed here can be applied to a wide range of arithmetic sequence problems, making this a valuable skill for anyone studying mathematics. Mastery of these concepts provides a solid foundation for further exploration of sequences, series, and other related mathematical topics.