Sequences
- View sequences as functions on ; connect discrete steps to continuous ideas.
- Understand convergence through – intuition and classic test cases.
- Meet arithmetic/geometric patterns, recurrence relations, and real-world growth/decay models.
- Practice-driven: short exercises with explanations in each chapter.
Every sequence hides a rule—like footsteps toward an unseen destination. Don’t stare at a single term; look ahead to the limit.
Symbols Used in This Article
Symbol notation
| Symbol | Type | Pronunciation/Explanation | Meaning in This Article |
|---|---|---|---|
| Mathematical symbol | natural numbers | The set of positive integers used as indices | |
| Mathematical symbol | Real numbers | The set of real numbers, the range of the terms | |
| Sequence notation | a sub n | The th term | |
| Sequence notation | sequence a_n | The whole sequence |
A sequence is the closest thing to watching change in discrete steps: the variable moves forward one integer at a time instead of sliding continuously. This course starts from the basic concept, works through the general terms and sums of arithmetic and geometric sequences, then the clever summation techniques — subtract reversed terms, group, and split the term — the method of mathematical induction as a basic proof tool, and finally recurrence relations and the criteria that decide when a sequence converges. It closes with applications to compound interest, loan repayment, population growth, and geometric problems.