Sequences: Concept, General Term and Monotonicity
A sequence is an ordered list of numbers a1, a2, a3, … in which each term is given by its position n, and the general term an is a formula or a rule that produces the nth term. In brief: the sequence with general term an = 2n + 1 is 3, 5, 7, 9, …, and the sequence an = 1/n is 1, 1/2, 1/3, 1/4, …. A sequence is increasing if an+1 > an for all n, and decreasing if an+1 < an.
A sequence is a function whose domain is the positive integers. Sequences describe repeated processes such as savings from month to month, populations from year to year and the steps of an algorithm. This guide explains the definition, the ways to describe a sequence (explicit formula, recursion, list), how to test monotonicity and boundedness and gives a table of the standard types, with worked examples.
What is a sequence?
A sequence is a function a: ℕ → ℝ, written (an), where an is the nth term. Unlike a set, the order matters and terms may repeat: the sequence 1, −1, 1, −1, … has an = (−1)n+1. A sequence can be finite (with a last term) or infinite. In an infinite sequence there is a term for every positive integer n, and the question of what happens as n grows is central.
How can a sequence be described?
- Explicit formula: an is given directly by n, such as an = n2, which gives 1, 4, 9, 16, …. Any term can be computed at once: a100 = 10,000.
- Recursive formula: the first term(s) and a rule connecting a term with the previous ones, such as a1 = 2, an+1 = 3an − 1, giving 2, 5, 14, 41, …. The Fibonacci numbers are defined by F1 = F2 = 1 and Fn+1 = Fn + Fn−1: 1, 1, 2, 3, 5, 8, 13, …
- Verbal description or list: “the prime numbers” 2, 3, 5, 7, 11, … has no simple formula.
Example 1: finding the general term from a pattern
The terms 4, 7, 10, 13, … increase by 3, so an = 3n + 1. The terms 2, 4, 8, 16, … are powers of 2: an = 2n. The terms 1, −4, 9, −16, … have alternating signs and squares: an = (−1)n+1n2. The terms 1/2, 2/3, 3/4, … give an = n/(n + 1).
Example 2: from a recursion to a formula
a1 = 5 and an+1 = an + 4 gives 5, 9, 13, …, so an = 4n + 1, an arithmetic sequence.
How do you test monotonicity?
A sequence is increasing if an+1 > an for all n, decreasing if an+1 < an, and non-decreasing or non-increasing if the inequalities are not strict. A sequence that is neither is not monotonic. Methods:
- Difference: compute an+1 − an and determine its sign.
- Ratio (for positive terms): compute an+1/an and compare with 1.
- Function: if an = f(n) and f is increasing on [1, ∞), then the sequence is increasing.
Example 3
an = n/(n + 1). Then an+1 − an = (n + 1)/(n + 2) − n/(n + 1) = [(n + 1)2 − n(n + 2)]/((n + 1)(n + 2)) = 1/((n + 1)(n + 2)) > 0. The sequence is increasing: 1/2, 2/3, 3/4, …
Example 4
an = 2n/n!. The ratio an+1/an = 2/(n + 1). For n = 1: 1 (so a2 = a1 = 2), and for n ≥ 2 the ratio is less than 1. The sequence is decreasing from the second term: 2, 2, 4/3, 2/3, 4/15, …
Example 5: not monotonic
an = (−1)n/n gives −1, 1/2, −1/3, 1/4, …, which alternates, so it is neither increasing nor decreasing. Another example: an = n2 − 6n has the terms −5, −8, −9, −8, −5, 0, …, decreasing for n ≤ 3 and increasing afterwards. The vertex of the parabola at n = 3 explains the change.
What is a bounded sequence?
A sequence is bounded above if there is a number M with an ≤ M for all n, bounded below if an ≥ m for all n, and bounded if both hold. For example, an = n/(n + 1) is bounded: 1/2 ≤ an < 1. The sequence an = n2 is bounded below by 1 but not above. A fundamental theorem says that a monotonic and bounded sequence converges to a limit.
Worked examples
Example 6: the first terms
an = (2n − 1)/(n + 3): a1 = 1/4, a2 = 3/5, a3 = 5/6, a4 = 7/7 = 1, a10 = 19/13. The terms increase towards 2.
Example 7: which term equals a value?
For an = 3n − 7, which term is 50? 3n − 7 = 50 gives n = 19. Is 25 a term? 3n − 7 = 25 gives n = 32/3, not an integer, so no.
Example 8: a recursion with two terms
b1 = 1, b2 = 3 and bn+2 = bn+1 + 2bn: b3 = 3 + 2 · 1 = 5, b4 = 5 + 2 · 3 = 11, b5 = 11 + 2 · 5 = 21, b6 = 43. This is a Jacobsthal sequence, and it has the closed form bn = (2n+1 − (−1)n+1)/3. Check: b1 = (4 − 1)/3 = 1, b2 = (8 + 1)/3 = 3 and b4 = (32 + 1)/3 = 11 ✓.
Table of basic sequences
| General term | First terms | Monotonicity | Bounded? |
|---|---|---|---|
| an = n | 1, 2, 3, 4, … | increasing | below only |
| an = 1/n | 1, 1/2, 1/3, 1/4, … | decreasing | yes |
| an = (−1)n | −1, 1, −1, 1, … | not monotonic | yes |
| an = 2n | 2, 4, 8, 16, … | increasing | below only |
| an = (1/2)n | 1/2, 1/4, 1/8, … | decreasing | yes |
| an = n/(n + 1) | 1/2, 2/3, 3/4, … | increasing | yes |
| an = (−1)n/n | −1, 1/2, −1/3, … | not monotonic | yes |
Where do sequences come from?
Number sequences are ancient: the triangular and square numbers were studied by the Pythagoreans, and the Fibonacci sequence appeared in Leonardo of Pisa’s Liber Abaci in 1202 in a problem about the growth of a rabbit population (though the sequence was known earlier in Indian mathematics). The modern concept of a sequence as a function on the natural numbers and of its limit was formalised in the nineteenth century with the arithmetization of analysis.
Where are they used today?
Computer programs use sequences when they iterate. Financial models use them for monthly balances, and in biology, sequences describe population counts per generation. Sequences defined by recursion, such as an+1 = (an + 2/an)/2 starting from 1, produce 1, 1.5, 1.4167, 1.41422, …, approximating √2 with increasing accuracy; this is Newton’s method.
Which mistakes are common?
Confusing a sequence with a set
The order and repetitions matter in a sequence.
Starting the index at the wrong value
an = 2n + 1 with n starting from 1 gives 3, 5, 7. With n starting from 0 it gives 1, 3, 5. State the start.
Testing monotonicity with only a few terms
A few terms can mislead. Prove the sign of an+1 − an for all n.
Assuming a pattern continues
Many formulas fit the first few terms. The list 1, 2, 4, 8, … could continue with 16 (powers of 2) or with 15 (a different rule).