1.61 Arithmetic Sequence Definitions
An arithmetic sequence is a series of numbers where each term increases by a constant difference, forming a foundational concept in algebra.
Arithmetic Sequence Definitions establishes the vocabulary for an ordered list of numbers that changes by the same fixed amount from one term to the next, describing that fixed amount specifically, and the numbering system used to identify each term's position within the list.
Arithmetic Sequence
An arithmetic sequence is an ordered list of numbers in which each term after the first is obtained by adding a fixed constant to the preceding term, such as 3, 7, 11, 15, 19, where each term is 4 more than the one before it. An arithmetic sequence may be finite, containing a limited number of terms, or infinite, continuing according to the same fixed pattern without end.
Arithmetic Common Difference
The arithmetic common difference is the fixed constant added to each term of an arithmetic sequence to obtain the next term, calculated by subtracting any term from the term immediately following it. The arithmetic common difference remains identical between every pair of consecutive terms throughout the entire sequence, and it is this constancy that defines the sequence as arithmetic in the first place; in the sequence 3, 7, 11, 15, 19, the arithmetic common difference is 4.
Arithmetic Sequence Index
The arithmetic sequence index is the whole number identifying a term's specific position within an arithmetic sequence, typically starting at 1 for the first term and increasing by 1 for each subsequent term, and commonly written as a subscript attached to the sequence's variable name, such as a₁, a₂, a₃, and so on. Once the first term and the arithmetic common difference are known, the arithmetic sequence index allows any term in the sequence to be located and computed directly, without needing to list every term that comes before it.
Together, these definitions describe the arithmetic sequence as an ordered list governed by a single fixed arithmetic common difference, with the arithmetic sequence index providing the positional labeling system that allows any specific term to be identified and computed directly from the sequence's starting value and common difference.