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} = a_{1} + d

Third Term: a _{3}= a_{2} + d = (a_{1} + d) + d = a_{1} + 2 d

Fourth Term: a _{4} = a_{3} + d = (a_{1} + 2d) + d = a_{1} + 3 d

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

N^{th} Term: a_{n}= a_{1} + (n - 1)d

The formula above is also known as the explicit formula and is used to determine the n^{th} term when the previous term is unknown. When simplified, this formula becomes: a_{n }= dn + (a_{1} - d) and islinear. The common difference is the slope and (a_{1} - d) is the y-intercept.