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# 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 *a*_{1}, ..., and the *n*th term by *a*_{n}. Then the terms up to and including the *n*th 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 *a*_{1} = −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 ...