In mathematics, a sequence is an ordered list of numbers or elements, where each element is referred to as a term of the sequence. Sequences can be finite or infinite, depending on whether they contain a limited number of terms or continue indefinitely.
A sequence can be defined by a specific rule or formula that describes how to generate its terms. For example:
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Arithmetic Sequence: A sequence where each term after the first is obtained by adding a constant difference, \( d \), to the preceding term. Example: \( 2, 5, 8, 11, \ldots \) (where \( d = 3 \)).
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Geometric Sequence: A sequence where each term after the first is obtained by multiplying the preceding term by a constant factor, \( r \). Example: \( 3, 6, 12, 24, \ldots \) (where \( r = 2 \)).
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Fibonacci Sequence: A specific sequence where each term is the sum of the two preceding terms, starting with 0 and 1. The sequence looks like this: \( 0, 1, 1, 2, 3, 5, 8, 13, \ldots \).
Mathematically, a sequence can also be defined by a function \( a_n \) that assigns a term to each natural number \( n \). The \( n \)-th term of the sequence is denoted as \( a_n \).
Overall, sequences are fundamental in various areas of mathematics and are used in analysis, number theory, and many applications in science and engineering.