How do you express the n-th term of an arithmetic sequence?

In an arithmetic sequence, each term after the first is generated by adding a constant difference, denoted as (d), to the previous term. The first term of the sequence is usually represented by (a_1).

The n-th term of an arithmetic sequence can be formulated as follows: start with the first term (a_1) and add the product of the difference (d) and the number of intervals (or steps) from the first term to the n-th term. Since there are (n-1) intervals between the first term and the n-th term, the formula for the n-th term is:

[ a_n = a_1 + (n-1)d ]

This formula captures the essence of how arithmetic sequences work: each term is derived by repeatedly adding the common difference (d) to the initial term.

This understanding is integral for solving problems that involve arithmetic sequences, as it allows one to determine any term in the sequence given the first term and the common difference.