Sequences and Series - Mathematics (Undergraduate Foundation)

This course provides a complete treatise of mathematical progressions. It formally defines sequences and series, with a specific focus on Arithmetic Progressions (APs) and Geometric Progressions (GPs). This material is the foundation for analysing patterns of growth, decay, and summation. These principles are foundational to finance, computer science, and physics. The course provides the tools to calculate compound interest, analyse algorithm complexity, and model physical phenomena. Mastery is required for calculus and financial mathematics. By the end of this course, you will be able to differentiate between sequences and series, apply all standard formulas for the nth term and sum of Arithmetic and Geometric Progressions, calculate the sum to infinity for a convergent GP, and solve problems involving arithmetic and geometric means. This course is designed for first-year university students in mathematics, economics, finance, and engineering. It is a mandatory prerequisite for the study of calculus and provides an essential foundation for financial modelling and algorithm analysis.

Enrolment valid for 12 months

Course Chapters

1. Introduction
3
This chapter provides a direct overview of the course. It defines the core concepts of sequences and series and establishes the notational conventions used to describe them, forming the logical basis for all subsequent topics. Key objectives include understanding the course structure, differentiating between a sequence and a series, and mastering the use of sigma notation for representing series.
Concept Overviews
3 Lessons
41:10
2. Arithmetic Progressions
4
4
This chapter provides a complete analysis of Arithmetic Progressions (APs). It covers the core formulas for determining any term in a sequence and for calculating the sum of a series, concepts fundamental to modelling linear growth. Key topics include the formulas for the nth term and the sum of n terms, and the arithmetic mean. Worked examples demonstrate solving for all unknown variables within a progression.
Concept Overviews
4 Lessons
32:15
Problem Walkthroughs
4 Lessons
28:40
3. Geometric Progressions
4
4
This chapter provides a complete analysis of Geometric Progressions (GPs). It details the formulas for the nth term and the sum of a series, which are essential for modelling exponential growth and decay processes, such as compound interest. Key topics include the formulas for the nth term, the sum of n terms, the geometric mean, and the sum to infinity for convergent series. Worked examples cover all standard problem types.
Concept Overviews
4 Lessons
32:59
Problem Walkthroughs
4 Lessons
22:16
4. Conclusion
2
This chapter consolidates the core concepts of the course. It provides a structured summary of arithmetic, geometric, and other progressions, reinforcing the essential principles required for progression. The conclusion summarises key definitions and formulas. It also outlines how these concepts will be applied in subsequent courses on calculus and financial mathematics.
Concept Overviews
2 Lessons
5:45
5. Other Notable Progressions
2
4
This chapter extends the study of progressions beyond arithmetic and geometric types. It introduces harmonic progressions and the formulas for summing the powers of natural numbers, which are specialised but important tools in analysis. Key topics include the definition of harmonic progressions and their relationship to APs, and the standard formulas for the sum of the first n integers, squares, and cubes.
Concept Overviews
2 Lessons
Problem Walkthroughs
4 Lessons