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An arithmetic sequence is a series of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference and is denoted as 'd'. One of the simplest examples of an arithmetic sequence is the natural numbers 1, 2, 3, 4, 5, and so on, where the common difference is 1.

In the context of your textbook exercise, the sequence starts with an initial term, \(a_1\), which is 3, and each subsequent term is found by subtracting 6 from the previous term. Here, the common difference \(d\) is -6, indicating that the sequence is decreasing by 6 each time. Understanding the nature of the arithmetic sequence and identifying the common difference are critical first steps in grasping the overarching concept of sequence patterns in arithmetic progressions.

In mathematics, being able to recognize patterns in sequences is an invaluable skill. Sequence patterns help you predict subsequent numbers and formulate rules that define the entire sequence.

For arithmetic sequences, this pattern is linear, and the sequence either increases or decreases at a steady rate, which is the common difference. Once you identify this pattern, you can describe the entire sequence using an explicit formula. In our exercise, the pattern is a repetitive decrease by 6, which is a clear indicator of an arithmetic sequence with a negative common difference. Grasping this pattern is essential for creating the explicit formula that expresses the sequence in terms of any term number, \(n\).

The explicit formula for an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\), where \(a_n\) represents the nth term in the sequence, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. The formula is designed to find any term without having to calculate all the previous terms in the sequence.

In the exercise, substituting the first term \(a_1 = 3\) and the common difference \(d = -6\) into the formula gives us \(a_n = 3 + (n-1)(-6)\). Simplifying this expression, we reach the rule that defines our sequence explicitly: \(a_n = 9 - 6n\). By using this rule, you can now calculate any term in the sequence directly. Understanding how to construct and simplify the arithmetic sequence formula is crucial for solving problems related to arithmetic progressions efficiently.