An increasing sequence of positive integers is a **complete sequence** if every term can be written as a sum of the first term, using the first term at most once [1].

A couple of variations on this definition:

- Schissel [2] states that a complete sequence if every positive integer is a sum of “one or more distinct terms” in the sequence.
- Erdos & Graham [3] describe a complete sequence as one where “every

*sufficiently large*” natural number is a sum of distinct terms of the sequence. This particular definition is called a**weakly complete sequence**[4]. - Linz & Jones [5] define a
**r-complete sequence**, where every sufficiently large positive integer can be represented as the sum of*r or more distinct terms*from the sequence.

## Complete Sequence Examples

A simple example of a complete sequence is {1, 2, 3, 4, …}.

Some complete sequences are more challenging to spot. For example, {1, 2, 3, 4, 8, 12, 16, 20, 24, 28, …} meets the definition because we can represent positive integers in modulo 4 (an arbitrary positive integer, a, can always be written as *a* = *n* * *q* + *r*).

The prime numbers are a complete sequence (if you add 1).

The Fibonacci sequence {1, 1, 2, 3, 5, 8, …} is an example of a complete sequence. Here [2],

- f
_{l, 1}(1) = 1, f_{1, 1}(2) = 1, and - f
_{1, 1}(n) = f_{1, 1}(n – 1) + f_{1, 1}(n – 2) if n ≥ 3.

Removing a single number still leaves a complete sequence, although removing two numbers does not [6].

## References

[1] Earl, R. (2017). Towards Higher Mathematics: A Companion. Cambridge University Press.

[2] Schissel , E. (!987). Characterizations of Three Types of Completeness. Retrieved April 7, 2021 from: https://www.fq.math.ca/Scanned/27-5/schissel.pdf

[3] Erdos, P. & Graham, R. (1980). Old and New Problems and Results in Combinatorial Number Theory: Monographie Numero 28 de L’Enseignement Math&matique .

Lf Enseignement Mathematique de 1’Universite de Geneve.

[4] Fox, A. & Knapp, M. (2013). A Note on Weakly Complete Sequences. Journal of Integer Sequences.

[5] Linz, W. & Jones, E. (2016). r-Completeness of Sequences of Positive Integers. Retrieved April 7, 2021 from: https://www.emis.de/journals/INTEGERS/papers/q59/q59.pdf

[6]Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.

**CITE THIS AS:**

**Stephanie Glen**. "Complete Sequence" From

**CalculusHowTo.com**: Calculus for the rest of us! https://www.calculushowto.com/complete-sequence/

**Need help with a homework or test question? **With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!