nth

Nth, written as n-th, is a placeholder term used in mathematics and related fields to denote the element in a sequence at position n, where n is a positive integer. The letter n is commonly used for the index variable, and the term describing its position adapts to the context, such as the nth term, nth derivative, or nth root. The convention assumes an ordering of objects.

Examples include sequences. For an arithmetic sequence with first term a1 and common difference d, the nth term is a_n = a1 + (n−1)d. For a geometric sequence with first term a1 and ratio r, a_n = a1 r^{n−1}. These explicit formulas express the value of the nth term directly in terms of n.

Other uses include the nth derivative, denoted f^(n)(x) or D^n f(x), which represents the result of differentiating a function n times. The nth root, x^{1/n}, is the number whose nth power equals x. In both contexts, n is a positive integer, though sometimes zero is allowed in derivative notation when n=0 gives the original function.

In common language, nth is part of the ordinal suffix system formed with -th, and is used when the exact index is unspecified or general. In computing and discrete mathematics, phrases such as the nth prime or the nth Fibonacci number describe elements without naming n’s value. The meaning is determined by the surrounding context and the imposed indexing convention.