1. Number and Algebra

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Arithmetic sequences and series (1.2)

Use of the formulae for the nthn^{th}nth term and the sum of the first nnn terms of the sequence.

Use of sigma notations of sums of arithmetic sequences.

Applications

Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

Geometric sequences and series. (1.3)

Use of the formulae for the nthn^{th}nth term and the sum of the first nn n terms of the sequence.

1.2 Sequences and Series

Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. (1.6)

The symbols and notation for equality and identity.

Proof by mathematical induction (AHL 1.15)

Proof by contradiction

Use of a counterexample to show that a statement is not always true.

1.3 Proof and Reasoning

The binomial theorem: expansion of (a+b)n(a + b)^n(a+b)n,  n∈Nn ∈ Nn∈N. (1.9)

Use of Pascal’s triangle and nCr^nC_rnCr​.

Counting principles, including permutations and combinations. (AHL 1.10)

Extension of the binomial theorem to fractional and negative indices, ie (a+b)n(a + b)^n(a+b)n, n∈Qn ∈ Qn∈Q.

Use of Pascal’s triangle

Figure 1.4.1 Pascal’s triangle

1.4 Binomial Theorems and Combinatorics

The number iii, where i2=−1i^2=-1i2=−1. (1.12)

Cartesian form z=a+biz = a + biz=a+bi ; the terms real part, imaginary part, conjugate, modulus and argument.

The complex plane.

Modulus–argument (polar) form: z=r(cos⁡θ+isin⁡θ)=rcisθz = r(\cosθ + i\sinθ) = rcisθz=r(cosθ+isinθ)=rcisθ (1.13)

Euler form: z=reiθ z = re^{iθ}z=reiθ

Sums, products and quotients in Cartesian, polar or Euler forms and their geometric interpretation.

Complex conjugate roots of quadratic and polynomial equations with real coefficients. (1.14)

1.5 Complex Numbers (HL)

1.6 Linear Equations (HL)