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Understanding the common difference is key when studying arithmetic sequences. It is essentially the steady increment or decrement between consecutive terms in the sequence. In other words, if you pick any term in the sequence and subtract its predecessor, you get the common difference.

In the given example, the common difference, denoted as \( d \), is calculated by subtracting the term at \( n=6 \) from the term at \( n=7 \). You end up with \( d = g(7) - g(6) = 12 - 8 = 4 \). So, the common difference for this particular sequence is 4.

This key feature helps define the nature of an arithmetic sequence - its predictability and linear pattern. By knowing the common difference, students can quickly discern the direction in which the sequence is heading and calculate subsequent terms with ease.

The sequence formula is a blueprint that enables us to calculate any term within an arithmetic sequence. For any arithmetic sequence, the standard formula is \( g(n) = a + d(n-1) \), where \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.

In our exercise, once we know the common difference is 4, the next step is to find the formula for the sequence, which involves determining the first term 'a'. By utilizing the known terms of the sequence, we can establish that \( a \) can be expressed in relation to \( g(6) \) and our common difference \( d \). After some rearrangement, we get \( a = g(6) - d(5) = 8 - 20 = -12 \).

This leads us to the sequence formula for this example: \( g(n) = -12 + 4(n-1) \). Such formulas provide a powerful tool allowing students to not only predict future terms but also back-calculate any term within the sequence.

Consecutive terms in an arithmetic sequence are pairs of terms that directly follow each other. They play a vital role because the common difference can always be found by subtracting one from another. The ability to work with consecutive terms is crucial for understanding and navigating through the sequence.

In practice, when you're given two consecutive terms like \( g(6) = 8 \) and \( g(7) = 12 \), you can identify the direction and rate at which the sequence is changing. This relationship between consecutive terms forms the foundation for solving many problems related to arithmetic sequences.

By simply subtracting the n-1 term from the nth term, or vice versa, you're leveraging the core principle that defines arithmetic sequences, making the concept of consecutive terms an indispensable part of learning sequence patterns.