Units 1 & 2 (Sequences and Series)
Develop fluency with arithmetic and geometric sequences, sigma notation, recursive definitions, and closed-form expressions. Explore convergence and divergence of series, model discrete growth/decay processes, and lay algebraic groundwork for calculus and combinatorics.

Unit 1: Foundations of Sequences and Recursion

1.1 Arithmetic Sequences and Series
Definition & Formulae

Applications:

• Modelling linear patterns

• Solving problems with evenly spaced data (e.g. finance, engineering)

1.2 Geometric Sequences and Series

Applications:

• Compound interest

• Exponential growth and decay modelling

1.3 Recursive Definitions

• Using recurrence relations to define sequences

• Translating between recursive and closed-form expressions

• Example: Fibonacci sequence and other difference-based patterns

1.4 Sigma Notation and Summation

• Interpreting and evaluating summations:

• Use of sigma notation in compact expression of series

Unit 2: Growth Patterns, Convergence, and Modelling

2.1 Convergence and Divergence of Series

• Criteria for convergence of infinite geometric series

• Behaviour of sequences

• Recognising divergent series and implications in applied settings

2.2 Applications of Series in Modelling

• Use of arithmetic/geometric series in real-world contexts

• Discrete models of depreciation, investments, and population growth

• Comparing recursive vs. closed-form for practical prediction

2.3 Introduction to Binomial Expansion (Prelude to Combinatorics)

• Binomial coefficients via Pascal’s Triangle

• Link to combinations

• Application in approximating functions and probability distributions

2.4 Technology Integration

• Use of TI-Nspire/Casio to compute and visualise sequences and series

• Graphical exploration of recursive patterns

• Verification of convergence/divergence behaviour numerically