Finite Sequence¶

Definition¶

A finite sequence is an ordered list with only finitely many entries, $a_m,\,a_{m+1},\,\ldots,\,a_n$ with integers $m\le n$ (usually $m=0$ or $m=1$$1$), written $(a_i)_{i=m}^n\text{.}$ It has $n-m+1$ terms. In contrast, a sequence $(a_n)_{n\ge m}$ never ends.

Interpretation¶

A finite sequence is an ordered list of numbers with a first and a last term: $a_m$ comes first, $a_n$ comes last, and each index in between carries exactly one term.

The figure shows $(a_i)_{i=0}^{5}$: one stick for each index $0,1,\dots,5$, so $5-0+1=6$ terms, and nothing after the last term $a_5$. The position along the axis is the index; the height of the stick is the value of the term.

Order and repetition both matter: $(1,2)$ and $(2,1)$ are different, and $(1,1,2)$ has three terms although the set $\{1,1,2\}=\{1,2\}$ has two elements. Counting terms is a common slip: $a_3,\dots,a_7$ has $7-3+1=5$ terms, not $4$. Later, the terms of a finite sequence are added up as a finite series.

Key Results¶

• Finite sequences have a last term.
• Every finite sequence can be viewed as a function on a finite index set.
• Finite sequences are always bounded.

Examples¶

• $(a_i)_{i=1}^3=(2,\,4,\,6)$ is a finite sequence.
• $(a_i)_{i=1}^1$ consists of a single term $a_1\text{.}$

Prerequisites¶

• Sequence

Related Concepts¶

• Bounded Sequence

Sources¶

• Math I: Calculus · Lecture 2

Last updated: 20 October 2026 03:02