Introduction to Arithmetic Sequences
Arithmetic sequences are a fundamental concept in mathematics, often appearing in various applications from basic arithmetic to more complex mathematical problems. This article explains how to find the first 10 terms of a given arithmetic sequence and provides a step-by-step process with detailed examples.
Understanding the Arithmetic Sequence
An arithmetic sequence is a sequence of numbers such that the difference between any two successive members is a constant. This constant difference is known as the common difference, denoted as d. The general form of an arithmetic sequence can be written as:
a, a d, a 2d, a 3d, ...
Expressing the Terms of an Arithmetic Sequence
The n-th term of an arithmetic sequence can be found using the formula:
a_n a_1 (n - 1)d
where a_1 is the first term of the sequence, d is the common difference, and n is the number of terms.
Example: First 10 Terms of the Sequence 4, 9, 14, 19
Given the sequence: 4, 9, 14, 19, we can identify the first term and the common difference.
Identifying the First Term and Common Difference
The first term, a_1, is 4.
The common difference, d, is calculated as:
d 9 - 4 5
Calculating the First 10 Terms
Using the formula for the n-th term, we can calculate the first 10 terms:
a_1 4 a_2 4 (2 - 1) times 5 9 a_3 4 (3 - 1) times 5 14 a_4 4 (4 - 1) times 5 19 a_5 4 (5 - 1) times 5 24 a_6 4 (6 - 1) times 5 29 a_7 4 (7 - 1) times 5 34 a_8 4 (8 - 1) times 5 39 a_9 4 (9 - 1) times 5 44 a_{10} 4 (10 - 1) times 5 49Thus, the first 10 terms of the sequence are:
4, 9, 14, 19, 24, 29, 34, 39, 44, 49
Conclusion
By understanding the formula for the n-th term of an arithmetic sequence and following the steps outlined in this guide, you can easily find any term in the sequence. The ability to identify and work with arithmetic sequences is valuable in many areas of mathematics and practical applications.
If you have any further questions or need additional clarification, feel free to ask!