In calculating the arithmetic sequence a_n = a_1 + 4d, what role does d play?

In the context of the arithmetic sequence given by the formula ( a_n = a_1 + 4d ), the variable ( d ) represents the common difference between successive terms of the sequence. In an arithmetic sequence, each term is formed by adding a constant value, known as the common difference, to the previous term.

For example, if the first term ( a_1 ) is known, the subsequent terms can be derived by continuously adding ( d ). The formula reveals that ( a_n ) is determined by the first term and the product of the common difference with the number of intervals (in this case, 4) that are added to the first term to obtain the ( n )-th term. Hence, the role of ( d ) is crucial because it dictates how much each term increases (or decreases) as you progress through the sequence.

This understanding highlights why the correct answer identifies ( d ) as the common difference in the arithmetic sequence.