a1a2an

a1a2an is a compact notational form used in mathematics and related fields to refer to a finite sequence of elements named a1, a2, ..., an. The exact interpretation of this string depends on the context: the a_i may be numbers, symbols from an alphabet, or more general objects. In many discussions, a1, a2, ..., an are treated as distinct items in a sequence or tuple.

In algebra and number theory, juxtaposition of symbols commonly denotes multiplication. Therefore a1a2...an can represent the product of the numbers a1, a2, ..., an, that is a1 × a2 × ... × an. In contrast, in combinatorics, formal languages, and computer science, the same notation frequently denotes concatenation: if the a_i are symbols from an alphabet, then a1a2...an is a word of length n formed by writing the symbols in order.

Examples help illustrate the dual usage. If a1=2, a2=3, a3=4, then a1a2a3 equals 2×3×4 = 24 in a multiplicative context. If instead the a_i are letters, such as a1='x', a2='y', a3='z', then a1a2a3 is the string "xyz" in a concatenation context.

Commonly, the notation appears in explanations of sequences, products, and words. Its meaning is not fixed by the symbol itself; readers must rely on the surrounding mathematical or disciplinary context to determine whether juxtaposition denotes multiplication or concatenation, or simply represents a finite ordered list of elements.

See also: sequences, tuples, concatenation, product notation, words (formal languages).