# Definition:Term of Sequence

## Definition

The elements of a sequence are known as its **terms**.

Let $\sequence {x_n}$ be a sequence.

Then the **$k$th term** of $\sequence {x_n}$ is the ordered pair $\tuple {k, x_k}$.

### Index

Let $\sequence {x_n}$ be a sequence.

Let $x_k$ be the **$k$th term** of $\sequence {x_n}$.

Then the integer $k$ is known as the **index** of $x_k$.

## Also defined as

Some sources gloss over the fact that a sequence is a mapping and define the **terms** to be the elements of the range of the sequence:

*We call $x_n$ the*$n$th term*of the sequence.*- -- 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*: $\S 4.2$

- -- 1977: K.G. Binmore:

However, this simplistic treatment lacks the precision of the definition provided here, and $\mathsf{Pr} \infty \mathsf{fWiki}$ does not endorse it.

## Also known as

A **term of a sequence** is referred to by some sources as an **element of the sequence**.

Some sources refer to the **general term** of a **sequence** when expressing it by means of a rule.

## Sources

- 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.15$: Sequences: Definition $15.2$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.2$: Sequences - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**term**:**1.** - 1992: Larry C. Andrews:
*Special Functions of Mathematics for Engineers*(2nd ed.) ... (previous) ... (next): $\S 1.2$: Infinite Series of Constants - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**general term** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**general term** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next):**term**