We may define an arithmetic progression as a set of numbers in which each one after the first is obtained from the preceding one by adding a fixed number called the common difference. Suppose we denote the common difference of an arithmetic progression by d, the first term by a1, ..., and the nth term by an. Then the terms up to and including the nth term can be written as
If Sn denotes the sum of the first n terms of an arithmetic progression, then
If the n terms on the right-hand side of Equation 1.2 are written in reverse order, then Sn can also be expressed as
Upon adding Equations 1.2 and 1.3, we obtain
EXAMPLE 1.1 Given the arithmetic progression −3, 0, 3, ..., determine the 50th term and the sum of the first 100 terms. For a1 = −3, the second term (0) minus the first term is 0 − (−3) = 3 = d, the common difference. Then, from Equation 1.1,
and, from Equation 1.4,
A geometric progression is ...