In mathematics, a sequence is an ordered list of objects in which repetitions are allowed and order matters. Sequences can be finite or infinite, depending on whether the list goes on forever or not. A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. Sequences are also of interest in their own right, and can be studied as patterns or puzzles, such as in the study of prime numbers.
In particular, a sequence is formally defined as a function whose domain is an interval of integers. This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences. Sequences can be arithmetic, geometric, or harmonic, depending on the relationship between the terms. An arithmetic sequence is one in which every term is created by adding or subtracting a definite number to the preceding number. A geometric sequence is one in which every term is obtained by multiplying or dividing a definite number to the preceding number. A harmonic sequence is one in which every term is the reciprocal of a number in an arithmetic sequence.
