Growth Curve Modeling: Theory and Applications

1

MATHEMATICAL PRELIMINARIES

1.1 ARITHMETIC PROGRESSION

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 a_n. Then the terms up to and including the nth term can be written as

If S_n 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 S_n can also be expressed as

Upon adding Equations 1.2 and 1.3, we obtain

or

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,

1.2 GEOMETRIC PROGRESSION

A geometric progression is ...