Indexth

Indexth is a term encountered in discussions of indexed sequences and data structures. It denotes the element located at a specified index within an ordered collection. The term is formed from index, meaning a position, together with the ordinal suffix -th, analogous to i-th or n-th. In formal writing, indexth(S, i) is used to indicate the value at position i in the sequence S, assuming a defined indexing convention.

In mathematical notation, a finite sequence is written as S = (a1, a2, ..., an). Under 1-based indexing, indexth(S, i) equals ai for i in {1, ..., n}. In programming languages that employ 0-based indexing, the corresponding operation is typically written as S[i], with i ranging from 0 to n−1. The term indexth therefore helps discuss retrieval abstractly, independent of the concrete syntax.

Examples: let S = [5, 8, 13, 21]. Then indexth(S, 3) equals 13 under 1-based indexing, while in 0-based indexing the same idea would be indexth(S, 2) = 13.

Complexity and retrieval: In array-like structures, indexth retrieval is typically O(1). In linked lists, it is O(n) because traversal is needed. In functional languages, indexth may be defined recursively as indexth(S, i) = head(S) when i = 1, otherwise indexth(tail(S), i − 1). The operational meaning remains the same: selecting the element at a specified position.

See also: nth element, i-th, indexing, 0-based vs 1-based indexing, and order statistics.