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In an arithmetic sequence, the common difference is a crucial element. It's what defines the sequence as arithmetic rather than random. The common difference is the amount you add (or subtract) from any term in the sequence to get to the next one.
For example, in the sequence 112, 115, 118, 121, each term increases by 3. This constant increase is called the common difference.
The formula to find the common difference \(d\) between terms is:

• If the sequence is \(a_1, a_2, a_3, \ldots\), then \(d = a_2 - a_1\).

If every pair of terms has the same difference, you have correctly identified the common difference. This consistency tells us we are indeed looking at an arithmetic sequence! Understanding the common difference allows you to explore further down the sequence and easily predict upcoming numbers.

The pattern of a sequence is the order in which numbers or terms are arranged, following a specific rule. Recognizing this pattern helps to understand or reconstruct an arithmetic sequence.
In our given sequence 112, 115, 118, 121, the rule or pattern is simple: add 3 to each term to arrive at the next one.
Here's how you can think about it:

• Start with 112.
• Keep adding 3 to find the next terms: 112 to 115, 115 to 118, and so on.

Recognizing this systematic pattern allows you to extend the sequence confidently. Identifying such regularities makes sequences predictable and easier to work with. When the pattern is clear, it not only helps in solving the problem at hand but also aids in understanding the larger concept of sequences in mathematics.

Predicting the next terms in an arithmetic sequence becomes straightforward once the common difference and pattern are understood.
With the sequence 112, 115, 118, 121, where the common difference is 3, we can predict future terms by continuing the pattern: add 3 to the most recent term.
Let’s walk through the predictions for the next three terms:

• From 121 (last term given), add 3 to get 124.
• Add 3 to 124 to get 127.
• Add 3 to 127 to get 130.

This systematic approach to predicting terms lets you extend sequences as far as needed. Whenever you have an arithmetic sequence, using the common difference to predict future terms is a reliable method that works every time.