Finite Sequence

Sequence and Series >

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(x² + 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:

a₁, a₂, a₃, a₄, a₅,… a_n.

The “n” in a_n is called the length of the string, the nth 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 = n² – 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 = n² – n + 12. Placing our inputs (1, 2, 3) into the formula gives:

• a_n = 1² – 1 + 12
• a_n = 2² – 2 + 12
• a_n = 3² – 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