## What is a Finite Sequence?

A sequence is a **list of ordered items** (usually numbers) which can repeat; If the list ends (in other words, if you can count all of the items) then it is called a **finite sequence **or *string*. For example, the sequence (1, 2, 3) has three numbers with a beginning and end, so it is a finite sequence.

The items in the sequence are called *elements*, *terms*, or *members*.

If you see a “…” at the end of a list, it’s an infinite sequence (meaning that it goes on and on until infinity):

- Finite sequence: (1, 2, 3)
- Infinite sequence: (1, 2, 3…)

Finite series are functions and can be formed with generating functions. For example, the finite sequence (6, 26, 66) is generated by the function [x(x^{2} + 4x + 1)].

## Formal Definition of a Finite Sequence

More formally, a finite sequence is defined as a sequence with a domain consisting of the set {1, 2, 3, … n}—the first n positive integers [1]. In other words, a finite sequence is any sequence that has the form:

a_{1}, a_{2}, a_{3}, a_{4}, a_{5},… a_{n}.

The “*n*” in a_{n} is called the length of the string, the *n*th term of the sequence, or the image of the integer *n* [2].

## Listing Terms of a Finite Sequence

**Example question:** Find all terms for the finite sequence:

a_{n} = n^{2} – n + 12 for 1 ≤ n ≤ 3

Step 1: **Figure out what your inputs are.**

Let’s take a look at the right hand side (1 ≤ n ≤ 3) before tackling this problem. The “n” and “a_{n}” are the inputs and outputs of the sequence; They are equivalent to the x and y you would find in a function. The expression “1 ≤ n ≤ 3” is saying that your input (n) must be greater than or equal to 1 and less than or equal to 3. So our inputs are:

1, 2, 3.

Step 2: **Put your inputs (from Step 1) into the formula. **Our formula is a_{n} = n^{2} – n + 12. Placing our inputs (1, 2, 3) into the formula gives:

- a
_{n}= 1^{2}– 1 + 12 - a
_{n}= 2^{2}– 2 + 12 - a
_{n}= 3^{2}– 3 + 12

Step 3: **Solve**.

- a
_{n}= 1 – 1 + 12 = 12 - a
_{n}= 4 – 2 + 12 = 14 - a
_{n}= 9 – 3 + 12 = 18

Step 4: **List your numbers (from Step 3) enclosed by parentheses.** The solution is:

(12, 14, 18).

**Next:** Finite Geometric Sequence

## References

[1] College Algebra: Tutorial 54A: Sequences. Retrieved April 4, 2021 from: https://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut54a_seq.htm

[2] Sequences. Retrieved April 4, 2021 from: http://www.csc.villanova.edu/~japaridz/Archive/1300/lect1.7/tsld001.htm