Topic 1 – Core: Algebra

Sequences and Series

Working with Sequences and Series

  1. Definition and finding the nth term
  2. Recurrence relationships
  3. Sigma notation

Arithmetic Progressions

  1. Arithmetic progressions
  2. Sum of the first n terms
  3. Finding a and d given two terms
  4. Working with consecutive terms

Geometric Series

  1. Geometric series

Logarithmic and Exponential Functions

Exponential Functions and Logarithms

  1. Simplifying and expanding

The Exponential Function ex and Natural Log Functions

  1. The exponential function ex
  2. Sketching exponential graphs based on transformations
  3. The natural logarithmic function, ln x

Permutations and Combinations

Permutations and Combinations

  1. Permutations of n different items
  2. Permutations of r items from n different items
  3. Combinations of n different items
  4. Combining combinations

Sequences and Series

Binomial Expansion

  1. Binomial expansion
  2. Exam Questions - Binomial expansion, comparing coefficients
  3. Exam Questions - Binomial expansion, estimating a value
  4. Exam Questions - Binomial expansion, other

Proof by mathematical induction

Sum of Series

  1. Proof of the sum of the series ∑r
  2. Proof of the sum of the series ∑r²
  3. Proof of the sum of the series ∑r³
  4. Proof for other series
  5. Exam Questions - Sum of series

Various Types of Proofs

  1. Exam Questions - Divisibility and multiple tests
  2. Exam Questions - Recurrence relations

Complex Numbers

Complex Numbers

  1. Real and imaginary numbers
  2. Addition, subtraction and multiplying complex numbers and simplifying powers of i
  3. Division of a complex number by a complex number
  4. Argand diagrams
  5. Modulus and argument of a complex number
  6. Solving problems with complex numbers
  7. Square roots of a complex number
  8. Solving quadratic equations with complex roots
  9. Solving cubic equations
  10. Solving quartic equations
  11. Exam Questions - Complex numbers
  12. Exam Questions - Finding roots
  13. Modulus-argument form of a complex number
  14. Exponential Form (Euler's relation)
  15. Multiplication and division rules for mod and argument of two complex numbers
  16. Exam Questions - Further complex numbers
  17. Complex numbers specimen questions (mixed exam boards)

De Moivre’s Theorem

  1. De Moivre's theorem
  2. Exam Questions - Expressing sin(nθ) and cos(nθ) in terms of sinθ and cosθ
  3. Expressing sinnθ and cosnθ in terms of sin(kθ) and cos(kθ)
  4. nth roots of a complex number
  5. Exam Questions - nth roots of a complex number

Systems of Linear Equations

Solutions of systems of linear equations

  1. Solutions of systems of linear equations