Sequences and Series - Mathematics (Undergraduate Foundation)

Master the algebraic structures governing ordered lists of numbers and their discrete summations. This course moves rapidly beyond elementary pattern recognition to establish a rigorous foundation in the mechanics of real sequences and series. We investigate the precise definitions defining arithmetic and geometric progressions, establish formal notation for general terms, and execute the manipulation of finite and infinite series using standard sigma notation. The curriculum is focused strictly on the structural and algebraic properties required for advanced mathematical analysis. Solid competence in manipulating discrete numerical patterns is essential for quantitative professional fields. Financial analysts rely on the algebra of geometric series for modelling compound interest, loan repayments, and annuities. Computer scientists utilise sequence structures to define discrete data sets and analyse iterative loops. This framework is indispensable for modelling growth, decay, or discrete accumulation processes in business and STEM sectors with absolute precision. Upon completion, you will possess the skills to accurately identify and define sequences through their general term and recursive formulae. You will demonstrate competence in calculating the nth term and the sum of the first n terms for both arithmetic and finite geometric series. Furthermore, you will command the conditions for convergence in infinite geometric series and acquire the ability to manipulate sigma notation fluently, expanding and simplifying discrete summations efficiently. This course is designed for students entering undergraduate foundation programmes requiring strong algebraic bases, particularly in mathematics, finance, and computer science. It also serves as an intensive refresher for professionals returning to academia or shifting into roles demanding precise numerical structuring. Prior exposure to standard secondary school algebra is assumed; focus is placed strictly on the mastery and application of these core discrete definitions.

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.

MTH 101: Elementary Mathematics I - Algebra and Trigonometry

Master the foundational mathematical structures essential for success in quantitative undergraduate degrees and professional technical roles. This comprehensive learning track delivers a rigorous treatment of algebra and trigonometry, moving rapidly from fundamental set theory and real number operations to advanced topics including matrix algebra, complex numbers, and analytical trigonometry. You will establish the critical problem-solving framework required for advanced study in calculus, engineering mechanics, and data science. This programme is primarily designed for first-year university students in STEM disciplines requiring strong analytical bases, particularly engineering, physics, computer science, and economics. It also serves as an intensive, high-level refresher for professionals returning to academia or shifting into data-driven roles demanding precise numerical literacy and logical structuring. Prior competence in standard secondary school mathematics is assumed; focus is placed strictly on mastery and application of core definitions. Upon completion, you will possess the skills to construct rigorous logical arguments using set theory and mathematical induction, model complex relationships with functions and matrices, and analyze periodic systems using advanced trigonometry. You will demonstrate competence in solving diverse equation types, from quadratics to linear systems, and manipulating complex numbers in engineering applications. This track prepares you directly for the mathematical demands of second-year university studies and technical professional certification exams.

Course Chapters

1. Introduction

This chapter establishes the fundamental definitions and notation for ordered numerical lists and their summations. Technical precision here is vital for distinguishing between sequences and series before advancing to complex progressions. You will master defining formal sequence notation; distinguishing between finite and infinite structures; identifying individual terms via subscripts; and utilising sigma notation for compact series representation.

Concept Overviews

3 Lessons

2. Sequences

This chapter covers the algebraic mechanics of ordered numerical lists defined by functional rules. Mastery of these structures is essential for translating discrete patterns into precise mathematical models using both direct and step-by-step methods. You will master generating terms from explicit formulae; defining sequences through recurrence relations; deriving general rules from numerical patterns; and verifying term accuracy through algebraic substitution.

Concept Overviews

3 Lessons

Problem Walkthroughs

8 Lessons

3. Series

This chapter covers the algebraic mechanics of discrete summation and the rigorous application of sigma notation. Technical proficiency here is vital for condensing and expanding series, providing the foundational logic for calculating total accumulations in financial and physical models. You will master executing summations using sigma notation; applying linear properties to simplify complex sums; extracting individual terms from partial sums; and classifying series as convergent, divergent, or oscillatory.

Concept Overviews

4 Lessons

Problem Walkthroughs

3 Lessons

4. Arithmetic Sequence and Series

Govern the mechanics of linear numerical progressions defined by a constant common difference. You will master the algebraic derivation of nth terms and partial sums, providing the essential foundation for modelling uniform growth and arithmetic accumulation in financial and physical systems. You will master: calculating the general term using first term and common difference; resolving unknowns within arithmetic sequences; and executing the summation of finite arithmetic series using standard formulae.

Concept Overviews

4 Lessons

Problem Walkthroughs

2 Lessons

5. Finite Geometric Sequence and Series

Govern the mechanics of exponential progressions defined by a constant common ratio. You will master the algebraic derivation of nth terms and finite sums, establishing the essential framework for modelling compound growth and depreciation in financial and scientific contexts. You will master: calculating the nth term using common ratios; resolving unknown variables within geometric sequences; and executing the summation of finite geometric series using standard algebraic formulae.

Concept Overviews

4 Lessons

Problem Walkthroughs

2 Lessons

6. Infinite Geometric Series

Govern the mechanics of convergent geometric progressions where terms sum to a finite limit. You will master the algebraic conditions required for convergence, establishing the essential framework for modelling long-term economic stability and physical decay constants. You will master: identifying convergence through common ratio constraints; deriving the sum to infinity formula; and converting recurring decimals into rational fractions using geometric series.

Concept Overviews

2 Lessons

Problem Walkthroughs

2 Lessons

7. Other Sequences and Series

This chapter covers complex sequences and special series that fall outside standard linear or exponential models. You will learn to synthesise arithmetic and geometric properties to resolve hybrid patterns and use standard identities for power summations. You will master identifying harmonic progressions through reciprocals; calculating nth terms for arithmetic-geometric sequences; and applying formulae for the sums of natural numbers and their squares.

Concept Overviews

5 Lessons

Problem Walkthroughs

1 Lesson