firstterm

Firstterm is a term used in mathematics and related disciplines to denote the initial element of an ordered collection, such as a sequence, list, or series. In standard notation the first term is frequently written as a_1, though some conventions use a_0, depending on whether indexing starts at 1 or 0. The concept is fundamental because it anchors the computation of subsequent terms and partial sums.

In arithmetic and geometric progressions, the first term determines all later terms through the common difference or ratio. For example, an arithmetic sequence with a_1 = 3 and difference d = 4 yields 3, 7, 11, 15, ...; a geometric sequence with a_1 = 2 and ratio r = 3 yields 2, 6, 18, 54, ...

In series, the first term is the initial addend in a finite sum or an infinite series S = a_1 + a_2 + a_3 + ... If indexing starts at zero, the first term would be a_0, and the partial sums would adjust accordingly.

In recurrence relations and differential equations, an initial term or initial condition provides the starting value from which subsequent terms or functions are determined.

In computing and data structures, the first term can refer to the element at the lowest index, often stored at position zero in zero-based systems.

Terminology notes: Although the concept is widely used, the single-word form firstterm is uncommon in formal mathematical writing; most authors use “first term” or “initial term” to convey the idea. The term should not be confused with “leading term,” which has a distinct meaning in asymptotic analysis and algebraic contexts.