Finding The 100th Term In Arithmetic Sequence 5, 10, 15, 20
Finding specific terms within a sequence is a fundamental concept in mathematics, especially when dealing with arithmetic sequences. An arithmetic sequence, characterized by a constant difference between consecutive terms, provides a predictable pattern that allows us to calculate any term in the sequence without having to list out all the preceding terms. In this article, we will explore the step-by-step method to determine the 100th term in the arithmetic sequence 5, 10, 15, 20, ... , providing a comprehensive understanding of the underlying principles and calculations involved. This process not only solves this specific problem but also equips you with the knowledge to tackle similar problems involving arithmetic sequences.
Understanding Arithmetic Sequences
Before diving into the solution, it's crucial to grasp the basics of arithmetic sequences. An arithmetic sequence is a series of numbers in which the difference between any two consecutive terms is constant. This constant difference is known as the common difference, often denoted as 'd'. For instance, in the given sequence 5, 10, 15, 20, ..., the common difference is 5 (10 - 5 = 5, 15 - 10 = 5, and so on). Identifying this common difference is the cornerstone for solving any problem related to arithmetic sequences.
The general formula for the nth term (an) of an arithmetic sequence is expressed as:
an = a1 + (n - 1)d
Where:
- an is the nth term we want to find.
- a1 is the first term of the sequence.
- n is the position of the term in the sequence (e.g., for the 100th term, n = 100).
- d is the common difference between consecutive terms.
This formula is a powerful tool that allows us to directly calculate any term in the sequence, provided we know the first term, the common difference, and the term's position. Understanding this formula is essential for efficiently solving problems related to arithmetic sequences.
Identifying Key Components
To find the 100th term of the sequence 5, 10, 15, 20, ..., the first step is to identify the key components required by the formula an = a1 + (n - 1)d. We need to determine the first term (a1), the common difference (d), and the term number (n). This initial step is crucial as it lays the foundation for the subsequent calculations. A clear understanding of these components ensures accurate application of the formula and leads to the correct solution.
- First Term (a1): The first term of the sequence is the initial value in the series. In this case, the first term, a1, is clearly 5. This is the starting point from which the sequence progresses, and it's a fundamental element in our calculation.
- Common Difference (d): The common difference is the constant value added to each term to obtain the next term. To find the common difference, we subtract any term from its succeeding term. For example, 10 - 5 = 5, 15 - 10 = 5, and 20 - 15 = 5. Thus, the common difference, d, is 5. This constant difference is what defines the sequence as an arithmetic sequence and allows us to predict future terms.
- Term Number (n): The term number represents the position of the term we want to find in the sequence. In this problem, we are looking for the 100th term, so the term number, n, is 100. This value tells us how far into the sequence we need to calculate.
With these key components identified, we are now well-prepared to apply the arithmetic sequence formula and calculate the 100th term.
Applying the Formula
With the key components identified, we can now apply the formula for the nth term of an arithmetic sequence: an = a1 + (n - 1)d. Substituting the values we found in the previous step, we have a1 = 5, d = 5, and n = 100. Plugging these values into the formula allows us to calculate the 100th term directly. This step is the heart of the solution, where we translate our understanding of the sequence into a concrete numerical answer.
Let's substitute these values into the formula:
a100 = 5 + (100 - 1) * 5
Now, we simplify the expression step by step:
a100 = 5 + (99) * 5
a100 = 5 + 495
a100 = 500
Therefore, the 100th term in the arithmetic sequence 5, 10, 15, 20, ... is 500. This result demonstrates the power of the arithmetic sequence formula, allowing us to efficiently calculate terms far into the sequence without having to manually list out each preceding term.
Conclusion
In conclusion, we have successfully determined the 100th term in the arithmetic sequence 5, 10, 15, 20, ... using the formula an = a1 + (n - 1)d. By identifying the first term (a1 = 5), the common difference (d = 5), and the term number (n = 100), we were able to substitute these values into the formula and calculate the 100th term to be 500. This exercise highlights the importance of understanding arithmetic sequences and their properties. The ability to find specific terms within a sequence is a valuable skill in various mathematical contexts. Furthermore, the methodical approach of identifying key components and applying the appropriate formula can be extended to solve a wide range of problems involving arithmetic sequences. Understanding and applying this method will not only help in academic settings but also in practical scenarios where sequential patterns need to be analyzed.
By mastering the concepts and techniques discussed in this article, you can confidently approach and solve similar problems, solidifying your understanding of arithmetic sequences and their applications. This skill is a building block for more advanced mathematical concepts and is an essential part of any mathematical toolkit.
