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

This course is also part of the following learning track. You may join the track to gain comprehensive knowledge across related courses.

[NUC Core] MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the mathematics every scientist, engineer, and computer scientist must know. This track covers the NUC Core syllabus for MTH 101: Elementary Mathematics I. From set theory and sequences to trigonometry and complex numbers, you will gain the exact tools used in higher mathematics, algorithm design, data analysis, physics, and engineering. This programme is for first-year university students in mathematics, computer science, engineering, or physics, as well as serious learners preparing for undergraduate study. It also serves professionals who need a rigorous mathematical foundation for technical fields. By the end, you will handle formal set notation, progressions, quadratic models, combinatorics, induction proofs, binomial expansions, trigonometric analysis, and complex arithmetic. You will be equipped to solve real problems in science, engineering, computing, and finance, and ready for advanced study in calculus, statistics, and applied mathematics.

Course Chapters

1. Introduction

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.

Chapter lessons

1-1. Welcome

A direct statement on the course's purpose and structure. This lesson explains the importance of mathematical progressions in modelling patterns of growth, decay, and summation.

1-2. Defining a sequence

Formally defines a sequence as an ordered list of numbers. The lesson clarifies the concept of a term and the notation used to represent the general term of a sequence.

1-3. Defining a series

Defines a series as the sum of the terms in a sequence. It differentiates between finite and infinite series, a critical distinction for later concepts.

1-4. Sigma notation

Introduces sigma (Σ) notation as the standard, efficient method for representing a series. This lesson covers how to read and write expressions using this notation.

2. Arithmetic Progressions

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.

Chapter lessons

2-1. Definition and common difference

Formally defines an Arithmetic Progression. It establishes the concept of the common difference as the constant value added to each term to produce the next.

2-2. The nth term formula

Derives and explains the formula for finding the value of any term in an AP. This is a primary tool for analysing arithmetic sequences.

2-3. The sum of n terms

Covers the derivation and application of the formula for the sum of the first n terms of an arithmetic series.

2-4. The arithmetic mean

Defines the arithmetic mean of two numbers and its relationship to an Arithmetic Progression. This lesson also covers the insertion of multiple arithmetic means.

3. Geometric Progressions

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.

Chapter lessons

3-1. Definition and common ratio

Formally defines a Geometric Progression. It establishes the concept of the common ratio as the constant factor by which each term is multiplied to get the next.

3-2. The nth term formula

Derives and explains the formula for finding the value of any term in a GP. This is the primary tool for analysing geometric sequences.

3-3. The sum of n terms

Covers the derivation and application of the formula for the sum of the first n terms of a geometric series.

3-4. The geometric mean

Defines the geometric mean of two numbers and its relationship to a Geometric Progression. This lesson also covers the insertion of multiple geometric means.

3-5. Sum to infinity

Introduces the concept of a convergent geometric series. It derives and applies the formula for the sum to infinity, a critical concept for calculus.

4. Other Notable Progressions

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.

Chapter lessons

4-1. Harmonic progressions

Formally defines a harmonic progression (HP). It establishes the critical relationship between an HP and its corresponding arithmetic progression of reciprocals.

4-2. Series of powers of natural numbers

Introduces the standard formulas for the sum of the first n integers, the sum of their squares, and the sum of their cubes. These are important results in discrete mathematics.

5. Conclusion

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.

Chapter lessons

5-1. Summary of progressions

A concise review of the key definitions, properties, and formulas for Arithmetic, Geometric, and Harmonic Progressions. This lesson ensures all material has been consolidated.

5-2. Next steps in calculus

Explains how the principles of sequences and series, particularly infinite series, are a direct prerequisite for the study of calculus and advanced mathematical analysis.