In mathematics, a sequence is an ordered list of objects, and a number sequence is an ordered list of numbers following a particular pattern. Each element in a sequence is called a term, and the length of a sequence can be infinite.
There are several common types of number sequences, including:
An arithmetic sequence is a sequence in which the difference between each successive term remains constant. The general form of an arithmetic sequence is:
an = a1 + f × (n-1)
For example, the sequence 1, 3, 5, 7, 9, ... has a common difference of 2.
A geometric sequence is a sequence in which each successive term is the product of the previous term and a fixed, non-zero number (common ratio). The general form of a geometric sequence is:
an = a × r^(n-1)
For example, the sequence 1, 2, 4, 8, 16, ... has a common ratio of 2.
A Fibonacci sequence is a sequence in which each number following the first two is the sum of the two preceding numbers. The first two numbers can be 1 and 1 or 0 and 1, depending on the chosen starting point. The Fibonacci sequence is written as:
an = an-1 + an-2
For example, the Fibonacci sequence starts with 0 and 1: 0, 1, 1, 2, 3, 5, 8, 13, ...
Sequences have applications in various mathematical disciplines, including the study of functions, spaces, and other mathematical structures. They are particularly useful as a basis for series and have many real-world applications.