nthterm

The term “nth term” refers to the general term of a sequence, the expression that gives the value of the nth element for any positive integer n. It is commonly written as a_n or T_n, with the index n indicating position in the sequence. The starting index can be 1 or 0, depending on the convention used for a given sequence.

An nth term may be given explicitly, as a closed-form formula in n, or defined recursively, where each term depends on previous terms. An explicit formula provides a direct computation of a_n from n, while a recursive definition specifies a_n through relations involving earlier terms (for example, a_n = f(a_{n-1}, a_{n-2}, …)) together with initial conditions.

Examples:

- Arithmetic sequence: a_n = a_1 + (n−1)d, where a_1 is the first term and d is the common difference.

- Geometric sequence: a_n = a_1 r^{n−1}, where r is the common ratio.

- Fibonacci sequence: defined by a_1 = 1, a_2 = 1, and a_n = a_{n−1} + a_{n−2} for n ≥ 3; it has a recursive definition and a known closed-form expression (Binet’s formula) but no simple elementary form for all n.

In the context of series, the nth term refers to the nth element of the sequence being summed. A basic convergence test notes that if a_n does not tend to zero as n → ∞, the series ∑ a_n cannot converge, though this is only a necessary condition, not sufficient.

Common cautions include keeping track of the starting index and distinguishing between an explicit nth-term formula and a recursive definition.