Mathematics in Cybersecurity

6Sequences and Series

A sequence is an ordered list of numbers, where each number in the list is called a term of the sequence. The order of the terms of the sequence cannot be changed, and those terms may or may not follow some sort of an observable or describable pattern. Sequences can be finite, where the number of terms is limited, or infinite, where the number of terms goes on forever.

The general form of a sequence is:

where a_n represents the nth term of the sequence. The subscript n is called the index of the sequence, and it starts from 1 and goes up to infinity for an infinite sequence. There are times when, for the sake of description using a formula, the sequence may begin with term zero, represented as a₀. An example of this will be presented shortly.

When describing or referring to a sequence whose nth term is given by an, the notation used is either {a_n} or . The value of k is infinity if the sequence is infinite, and it is a constant if the sequence is finite.

6.1 Sequences

There are different ways to define a sequence. The most common are:

• Explicit Formula: Each term of the sequence is defined by a formula that depends on the index n as the independent variable. For example, the sequence 2, 4, 6, 8, … can be defined by the formula a_n = 2n. The formula to produce ...