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An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms remains constant. This difference is known as the "common difference." It is crucial because it defines the pattern of the sequence.

To find the common difference, you subtract any term from the term that follows it in the sequence. For example, in the sequence 6, 13, 20, 27, ..., the common difference (\(d\)) is 7. You can see this by calculating 13 - 6. This consistency in change is what forms the base of each step in an arithmetic sequence, guiding you from one term to the next.

Understanding the common difference helps you predict future terms in the sequence and is essential for calculating other properties, like the sum or specific terms, efficiently.

In an arithmetic sequence, the first term, often denoted as "\(a\)," is the starting point of the sequence. In our example, the first term is 6. The importance of the first term lies in its role as the anchor of the sequence.

The first term, combined with the common difference, allows you to build the entire sequence. If you know the first term, you have the base value from which all other terms are derived. Moreover, when calculating properties like the sum of a sequence, the first term is a crucial variable that helps determine the overall value.

In the arithmetic sequence sum formula, \(S_n = \frac{n}{2} (2a + (n - 1)d)\), the first term appears prominently, reiterating its fundamental importance in this type of sequence.

The number of terms in an arithmetic sequence is represented by \(n\) and indicates how many numbers are in the sequence. Knowing the number of terms is essential, especially when calculating sums or identifying specific terms in the sequence.

For the sequence given in the exercise, the number of terms is 14. This implies that there are 14 sequential numbers, starting from 6 and increasing by the common difference of 7 each time.

When you're calculating the sum of an arithmetic sequence, \(n\) helps determine the number of times the common difference needs to be applied. It defines the extent of the sequence and is integral for full comprehension, whether you're expanding the sequence or simplifying calculations with formulas like the sum of an arithmetic series.