Sequence (Mathematics)

Context: FIT1058_MOC · an ordered list, formalised as a function on $\mathbb{N}$ · contrast with an unordered set · the home of recurrences, limits and growth analysis

Quick Revision

• 🎯 Objective: an ordered list = a function on $\mathbb{N}$ ➔ $n$-th term $f_n$.
• 📦 Core Components: ordered + may repeat ➔ closed form vs recurrence.
• ⚡ Key Constraint: must be pinned by a rule, not an informal "$\dots$"; contrast the unordered set.

📝 Core

1. The Definition

• Sequence ➔ a function $f:\mathbb{N}\to A$ (infinite) or $f:[1,n{]}_{\mathbb{N}}\to A$ (finite).
• $n$-th term ➔ $f(n)=f_n$; number sequence when $A$ is a number set.
• Length ➔ size of a finite sequence’s domain.

2. Sequence vs Set

• Set ➔ unordered, no repetition.
• Sequence ➔ ordered, may repeat ($1,1,2,3,5$).
• Finite sequence ➔ an $n$-tuple; a string is a sequence of letters.

3. Defining a Sequence

• Not an "$\dots$" list ➔ only suggests a pattern.
• Closed form ➔ $f_n$ in $n$ alone ($n^2$).
• Recurrence ➔ $f_n$ from earlier terms + base case(s).

Key identities:

$\text{closed: }(n^2{)}_{n=1}^{\infty }=1,4,9,16,25,\dots$ $\text{recurrence: }{f}_1=1,f_n=n{f}_{n-1}(\text{factorials})$

⚖️ Core Decision Matrix

Feature	Sequence	Set
order	matters	irrelevant
repetition	allowed	none
formalism	function on $\mathbb{N}$	collection
finite form	$n$-tuple	—

When It Flips: sequences are fundamental to computation — memory is laid out in order, and time-vs-input-size is the sequence Big-O Notation analyses. Closed form gives any term in one step; a recurrence is often more natural but hides growth until unrolled.

📊 Exam Execution Trace

Applied Exercise

Problem: List the first five terms of $(n^2:n\in \mathbb{N})$ and state domain/codomain. Derivation Proof / Hand-Calculation Walkthrough:

$$\begin{array}{r}f:\mathbb{N}\to \mathbb{N},f(n)=n^2\\ f(\mathrm{1..5})=1,4,9,16,25,f(4)=16\end{array}$$

Final Extracted Output: an infinite sequence (function on $\mathbb{N}$); repetition allowed but here all distinct.

⚠️ Common Mistakes

• 💡 "$f_1,f_2,\dots$" is not a definition ➔ many sequences share any finite prefix; pin it with a closed form or a recurrence + base case.

🧠 Active Recall

Define a sequence precisely, and how it differs from a set.

• Hint: Ordered + repeatable.

Answer

• Short answer: A function $f:\mathbb{N}\to A$ (or $[1,n]$); ordered and may repeat, unlike a set.
• Why: $n$-tuple ➔ a finite sequence is a tuple; length = domain size.

Why is " $f_1,f_2,f_3,\dots$" not a definition, and what are the two valid ways?

• Hint: Rule required.

Answer

• Short answer: An "$\dots$" list only hints a pattern; define by closed form or recurrence + base case.
• Why: Ambiguity ➔ many sequences share any prefix.