What is the common difference in the arithmetic sequence defined by the formula a_n = a_1 + 4d?

In an arithmetic sequence, the general term can be expressed as a linear function of the first term and the common difference. The formula provided, ( a_n = a_1 + 4d ), highlights that the sequence is created based on a starting term ( a_1 ) and increments determined by ( d ).

Here, the term ( 4d ) signifies that the common difference does not remain constant for the entire sequence, but rather it is multiplied by 4 before being added to the first term. The common difference in an arithmetic sequence typically represents the consistent additive value between consecutive terms. In this case, if ( d ) is understood as that additive value, it indicates that each term is derived from the previous term by adding ( d ).

Thus, the common difference of the sequence is represented directly in ( d ), making it the foundational element that determines how the sequence progresses from one term to the next.