Sequences and Series - Mathematics (Undergraduate Foundation)

Master the algebraic rules governing ordered lists of numbers and their total sums. This course establishes a solid foundation in the mechanics of sequences and series. We investigate precise definitions for arithmetic and geometric progressions; use formal notation for general terms; and handle finite and infinite series using sigma notation. The curriculum focuses on the structural properties required for university-level mathematical analysis. Knowing how to handle numerical patterns is vital for many modern careers. Financial analysts use geometric series to calculate compound interest, loan repayments, and annuities. Computer scientists use sequence structures to organise data sets and analyse computer loops. This knowledge is necessary for measuring growth, decay, and accumulation in business and science with absolute accuracy. You will learn to identify and define sequences using general terms and recurrence formulas. You will calculate the nth term and the sum of terms for arithmetic, geometric, and harmonic progressions. You will master sigma notation; determine when infinite series converge; convert recurring decimals into fractions; and use formulas to sum natural numbers, squares, and cubes. This course is for students entering university foundation programmes in mathematics, finance, or computer science. It serves as an intensive refresher for professionals returning to academia or shifting into roles that demand precise numerical work. Even those outside these fields will benefit from the sharp logical thinking and problem-solving speed developed through pattern analysis.

15 hrs

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

36:48

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

56:54

Problem Walkthroughs

8 Lessons

31:53

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

1:00:04

Problem Walkthroughs

8 Lessons

32:45

4. Arithmetic Sequences and Series

This chapter covers arithmetic sequences where terms increase or decrease by a constant common difference. Mastering these linear patterns is essential for understanding uniform growth, simple interest, and steady accumulation in science and finance. You will master calculating general terms and sequence lengths; evaluating finite series sums; inserting arithmetic means; and using symmetric notation to solve algebraic progression problems involving ratios and unknown variables.

Concept Overviews

4 Lessons

1:13:10

Problem Walkthroughs

12 Lessons

1:23:53

5. Finite Geometric Sequences and Series

This chapter covers geometric sequences where terms change by a constant ratio. Mastering these exponential patterns is essential for modelling compound interest, asset depreciation, and physical rebound heights in science and finance. You will master calculating general terms and finite sums; inserting geometric means; and applying symmetric notation or logarithms to solve for unknown variables in real-world growth and decay scenarios.

Concept Overviews

4 Lessons

51:16

Problem Walkthroughs

13 Lessons

1:48:14

6. Infinite Geometric Series

This chapter covers infinite geometric series that sum to a finite limit. Mastering convergence is vital for limits, calculus, and professional financial modelling. You will define conditions for series convergence; calculate the sum to infinity; expand recurring decimals into geometric series; and convert these decimals into exact fractions.

Concept Overviews

2 Lessons

40:34

Problem Walkthroughs

5 Lessons

30:54

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

3 Lessons

53:09

Problem Walkthroughs

6 Lessons

40:38