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When we talk about arithmetic sequences, we are referring to a collection of numbers in which each number after the first is found by adding a consistent amount, known as the common difference, to the previous number. This common difference can be positive or negative. In the given exercise, the sequence is decreasing, which means the common difference is negative.

Let's illustrate this further using the provided sequence: 18, 15, 12, 9, ... Here, the common difference is -3, because each time we move from one number to the next, we subtract 3. Thus, an arithmetic sequence is essentially a series of numbers with a uniform subtraction (or addition) pattern. When spotting an arithmetic sequence, always seek to identify this consistent step – it is the key to understanding and predicting the behavior of the sequence.

Sequence prediction is a fundamental skill in mathematics. It allows us to identify the continuation of a numerical pattern based on its established behavior. As illustrated in the exercise, once the pattern within a sequence is recognized, it becomes possible to predict future numbers with accuracy.

For the sequence 18, 15, 12, 9, ..., predicting the next numbers involves understanding the rule that governs the sequence – in this case, subtracting 3. This exercise demonstrates a straightforward application of sequence prediction by consistently applying the subtraction pattern. It's like following a breadcrumb trail; each number leads you to the next, and the pattern acts as your guide. When attempting to predict the next values in an arithmetic sequence, don't just look at two adjacent numbers. Consider the overall progression and ensure that the pattern holds true for all the steps taken so far.

Subtraction patterns are integral to identifying arithmetic sequences that are decreasing. The pattern we see in the textbook problem is a perfect demonstration of a subtraction pattern, where each subsequent number in the sequence is smaller than the one before it.

The ability to quickly recognize and apply these patterns is crucial for solving sequence problems. Once the rule of subtraction by 3 is confirmed, it can be used to extend the sequence indefinitely. Practice with a variety of sequences can help sharpen this skill. Remember, the concept of maintaining the same difference—whether it's subtraction or addition—is what ties the sequence together and keeps it uniform, allowing for effective prediction and extension of the sequence.