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?

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 $\{a_n\}$ .

Function view

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

$\mathbb{N}$ （natural numbers）：正整数集合，是本章默认的索引集。
$\mathbb{R}$ （Real numbers）：实数集，常作为项的取值范围。

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., $1, \tfrac12, \tfrac13, \dots$

Link between general term and recurrence

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 $d$ .

Arithmetic general term

a_n = a_1 + (n-1)d

$d$ （difference）：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 $q$ .

Geometric general term

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

$q$ （quotient）：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

Bounded sequence

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

练习 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(3 个标签)

sequencesarithmetic sequencesgeometric sequences

$a_{n+1} - a_n = -3$ ⇒ arithmetic, $d = -3$ .
$\dfrac{b_{n+1}}{b_n} = -\tfrac12$ ⇒ geometric, $q = -\tfrac12$ .

练习 2

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

Reference Answer(3 个标签)

sequencesarithmetic sequencesgeometric sequences

Sum the recurrence: $c_{n+1} = 2 + \sum_{k=1}^{n} 2^{k} = 2 + (2^{n+1}-2) = 2^{n+1}$ , so $c_n = 2^{n}$ .

总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\mathbb{N}$	数学符号	natural numbers	索引的取值集合
$\mathbb{R}$	数学符号	Real numbers	项的取值集合
$\{a_n\}$	数列记号	sequence a_n	一个数列
$a_n$	元素符号	a sub n	第 $n$ 项
$d$	参数	difference	等差数列公差
$q$	参数	quotient	等比数列公比

中英对照

中文术语	英文术语	音标	说明
数列	sequence	/ˈsiːkwəns/	以正整数为索引的函数
通项公式	general term	/ˈdʒenərəl tɜːrm/	直接给出第 $n$ 项
递推公式	recurrence relation	/rɪˈkɜːrəns rɪˈleɪʃən/	通过前项生成后项
等差数列	arithmetic sequence	/ˌærɪθˈmetɪk ˈsiːkwəns/	相邻项差为常数
等比数列	geometric sequence	/ˌdʒiːəˈmetrɪk ˈsiːkwəns/	相邻项比为常数
有界	bounded	/ˈbaʊndɪd/	绝对值被同一常数约束
单调	monotonic	/ˌmɒnəˈtɒnɪk/	项随索引一致增或减

Next等差数列