An ordered list with a rule
A sequence is an ordered list of numbers, like 2, 4, 6, 8, 10, … . Each number in the list is a term. What makes a sequence different from a random pile of numbers is order and a rule: the position matters, and there is usually some pattern that tells you how to get from one term to the next.
We label terms by their position using subscripts: a₁ is the first term, a₂ the second, and in general aₙ is the n-th term, where n is a counting position — a natural number like 1, 2, 3, … . So for the list above, a₁ = 2, a₂ = 4, a₃ = 6, and a₅ = 10.
From a rule to a list
Often a sequence is given by an explicit formula — a formula for aₙ in terms of n. To get any term, you simply substitute the position n and evaluate. This is exactly the “plug in and compute” skill from earlier algebra, now applied to a counting variable.
Given the explicit formula a_n = 3n - 1, list the first four terms. a_1 = 3(1) - 1 = 3 - 1 = 2 a_2 = 3(2) - 1 = 6 - 1 = 5 a_3 = 3(3) - 1 = 9 - 1 = 8 a_4 = 3(4) - 1 = 12 - 1 = 11 Sequence: 2, 5, 8, 11, ...
The reverse skill — spotting the rule from a list — is the heart of pattern-finding. Look at the differences between consecutive terms, or their ratios. In 2, 5, 8, 11, … each term is 3 more than the one before, which hints at a rule built from 3n. We will name and use these patterns in the next two guides.
Finite, infinite, and indexing
A sequence can be finite (it stops, like 1, 4, 9, 16) or infinite (it continues forever, written with a trailing …, like 1, 4, 9, 16, …). The “…” means “keep following the same rule.”