Explicit Models

The following steps lead to an important discovery. Remember, when finding the next term in an arithmetic sequence, add the common difference, d.

First Term: a 1

Second Term: a 2 = a1 + d

Third Term: a 3 = a2 + d = (a1 + d) + d = a1 + 2 d

Fourth Term: a 4 = a3 + d = (a1 + 2d) + d = a1 + 3 d

* Notice that in each term, the coefficient of d is always one less than the position of the term: a4 = a1 + (4-1)d = a1 + 3d.

Nth Term: an
= a1 + (n - 1)d

The formula above is also known as the explicit formula and is used to determine the nth term when the previous term is unknown. When simplified, this formula becomes: an = dn + (a1 - d) and is linear. The common difference is the slope and (a1 - d) is the y-intercept.