Topic 1 covers number patterns and algebraic structures. At SL: arithmetic/geometric sequences, sigma notation, binomial theorem, and logarithms. HL extends to complex numbers, proof by induction, counting principles, and the full binomial theorem for any rational exponent.
Arithmetic sequence: uₙ = u₁ + (n−1)d; sum Sₙ = n/2(2u₁ + (n−1)d). Geometric sequence: uₙ = u₁ · rⁿ⁻¹; sum Sₙ = u₁(rⁿ − 1)/(r − 1). Infinite geometric series converges when |r| < 1: S∞ = u₁/(1 − r). Applications include compound interest, population growth, and loan repayments.
(a + b)ⁿ = Σ C(n,r) · aⁿ⁻ʳ · bʳ for r = 0 to n. SL covers positive integer n; HL extends to rational n using the generalised form. Students must find specific terms, coefficients, and apply to approximations.
Laws: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n log a. Change of base: logₐb = ln b / ln a. Solving exponential equations aˣ = b using logarithms. Natural logarithm ln and the number e.
z = a + bi; modulus |z| = √(a² + b²); argument arg(z) = arctan(b/a). Polar form z = r·cis(θ). De Moivre\'s theorem: [r·cis(θ)]ⁿ = rⁿ·cis(nθ). nth roots of complex numbers. Euler\'s form z = re^(iθ). Conjugate roots theorem for polynomial equations.
SL covers sequences, series, binomial theorem (positive integer n), and logarithms. HL adds: complex numbers (Cartesian, polar, Euler forms), proof by induction, counting principles (permutations, combinations), partial fractions, and infinite series convergence. HL Paper 3 may also test proof-based questions from this topic.
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