This is a beta course, so its structure, chapters, and examples may continue to change.

Basic Concepts of Sequences

Sequences act as functions on positive integers, bridging discrete steps to the continuous ideas of limits and calculus.

What Is a Sequence?

A sequence is a function whose independent variable is a positive integer. For each $n \in \mathbb{N}$ , the value is denoted $a_n$ , and the whole sequence is written $\{a_n\}$ .

a : \mathbb{N} \to \mathbb{R}, \quad a(n) = a_n

$\mathbb{N}$ (natural numbers): the set of positive integers, the default index set of this chapter.
$\mathbb{R}$ (real numbers): the set of real numbers, often used as the range of the terms.

Typical Examples

Natural numbers: $a_n = n$ (linear growth)
Harmonic: $a_n = \dfrac{1}{n}$ (used in limits/estimates)
Alternating: $a_n = (-1)^n$ (sign flips with period $2$ )

Ways to Represent Sequences

General term: give $a_n$ directly. Example $a_n = 2n + 1$ ⇒ $3, 5, 7, \dots$
Recurrence: generate $a_{n+1}$ from earlier terms. Example $a_1 = 1,\; a_{n+1} = 2a_n$ ⇒ $1, 2, 4, 8, \dots$
Listing: write early terms for intuition, e.g., $1, \tfrac12, \tfrac13, \dots$

If a recurrence $a_{n+1} = f(a_n)$ can be iterated, the closed form may be found by accumulating the relation. Conversely, a general term can often yield a recurrence by rearrangement.

Common Types

Arithmetic sequences

Consecutive terms differ by a constant $d$ .

a_n = a_1 + (n-1)d

$d$ (difference): the common difference of an arithmetic sequence.

Example: $2, 5, 8, 11, \dots$ with $d = 3$ . If $d>0$ the sequence increases; if $d<0$ it decreases.

Geometric sequences

Consecutive terms have a constant ratio $q$ .

a_n = a_1 \cdot q^{\,n-1}

$q$ (quotient): the common ratio of a geometric sequence.

Example: $3, 6, 12, 24, \dots$ with $q = 2$ . When $|q| < 1$ , $a_n \to 0$ , a classic convergence case.

Basic Properties

Boundedness

A sequence $\{a_n\}$ is bounded if $\exists M>0$ with $|a_n| \le M$ for all $n$ .

Example: $\left\{\tfrac{1}{n}\right\}$ is bounded; $\{n\}$ is unbounded.

Monotonicity

Increasing: $a_{n+1} \ge a_n$ for all $n$
Decreasing: $a_{n+1} \le a_n$ for all $n$

A sequence that is both monotonic and bounded converges—a key bridge to limits.

Periodicity and Parity

Periodic: $\exists T \in \mathbb{N}$ with $a_{n+T} = a_n$ (e.g., $(-1)^n$ with $T=2$ ).
Even/odd patterns: extending $\{a_n\}$ to a function can mirror function parity concepts.

Exercises

Exercise 1

Classify the sequence and give the parameter:

$a_n = 4 - 3n$
$b_n = 5 \cdot \left(-\dfrac{1}{2}\right)^{n-1}$

Reference Answer

sequencesarithmetic sequencesgeometric sequences

Approach: A sequence is arithmetic when consecutive differences are constant, and geometric when consecutive ratios are constant. Compute the difference for the first sequence and the ratio for the second, then read off the parameter.

Steps:

For $a_n = 4 - 3n$ : the difference is $a_{n+1} - a_n = (4 - 3(n+1)) - (4 - 3n) = -3$ , a constant. So the sequence is arithmetic with $d = -3$ .
For $b_n = 5 \cdot \left(-\dfrac{1}{2}\right)^{n-1}$ : the ratio is $\dfrac{b_{n+1}}{b_n} = -\dfrac12$ , a constant. So the sequence is geometric with $q = -\dfrac12$ .
Note that $q < 0$ means the terms alternate in sign and shrink in magnitude.

Answer: 1) arithmetic with $d = -3$ ; 2) geometric with $q = -\dfrac12$ .

Exercise 2

Sequence $\{c_n\}$ satisfies $c_1 = 2$ , $c_{n+1} = c_n + 2^{n}$ . Find a closed form.

Reference Answer

sequencesarithmetic sequencesgeometric sequences

Approach: The recurrence defines each term through a running sum of powers of two. Unroll it for a couple of terms first to spot the pattern, then evaluate the geometric sum in closed form.

Steps:

Unroll the recurrence: $c_1 = 2$ , $c_2 = 2 + 2 = 4$ , $c_3 = 2 + 2 + 4 = 8$ , which suggests $c_n = 2^n$ .
Prove it from the recurrence. Summing the recurrence gives $c_{n+1} = 2 + \sum_{k=1}^{n} 2^{k} = 2 + (2^{n+1} - 2) = 2^{n+1}$
Therefore $c_n = 2^{n}$ for every $n \geq 1$ , matching the values computed in step 1.

Answer: $c_n = 2^{n}$ .

Summary

Symbols Used in This Article

Symbol notation

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\mathbb{N}$	Mathematical symbol	natural numbers	The set of values used as indices
$\mathbb{R}$	Mathematical symbol	Real numbers	The set of values the terms take
$\{a_n\}$	Sequence notation	sequence a_n	A sequence
$a_n$	Element symbol	a sub n	The $n$ th term
$d$	Parameter	difference	The common difference of an arithmetic sequence
$q$	Parameter	quotient	The common ratio of a geometric sequence

Arithmetic Sequence

Found an issue with this content?

Tell us what could be improved.

Unlock Full Course ·

Your membership covers this course — no need to buy it again.

Login Required to Purchase

Payment Failed

Payment Successful!

Price History