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Understanding sequence notation is crucial when approaching problems involving sequences, such as finding a particular term in an arithmetic sequence. Sequence notation uses subscript numbers to identify the position of a term in a sequence. For instance, in the notation \( a_n \), \( a \) represents the term in the sequence, and \( n \) indicates the position of the term. This signifies that the \( n^{th} \) term can be found using a specific formula related to \( n \).

In arithmetic sequences, this notation helps us identify the position and value of each term in the sequence. For example, the first term is represented as \( a_1 \), the second term as \( a_2 \), and so on. With this systematic method, it becomes easier to discuss and calculate the terms within the sequence without ambiguity or confusion.

The 'common difference' in an arithmetic sequence is one of its defining features. It represents the constant value added to each term to obtain the next term in the sequence. Generally denoted by \( d \), it can be positive, negative, or even zero, which leads to an increasing, decreasing, or constant sequence respectively.

To determine the common difference, you simply subtract any term from the subsequent term. In our exercise example, the common difference is \( d=3 \), indicating that each term is 3 units greater than the preceding term. Understanding the common difference is essential because it serves as the building block for generating the terms of the sequence and for determining the arithmetic sequence formula.

The nth term calculation of an arithmetic sequence is a formula that allows one to find any term in the sequence without having to enumerate every preceding term. This formula is especially helpful when you need to identify a term that's far along in the sequence, such as the 100th term.

The generic formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n - 1) \cdot d \), where \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term's position in the sequence. To illustrate, with \( a_1 = 1 \) and \( d = 3 \), substituting these values into the formula yields the specific nth term formula for the given sequence: \( a_n = 1 + (n - 1) \cdot 3 \), which simplifies to \( a_n = 3n - 2 \). This final expression is a direct and efficient way to compute the value of any term in the sequence based on its position.