A-LevelFurther MathematicsComplex numbers9231/12 · May/June 2019 · Q810 Marks
Prove by mathematical induction that, for z ≠ 1 and all positive integers n, 1 + z + z² + ... + zⁿ⁻¹ = (zⁿ - 1)/(z - 1)
- (i)Prove by mathematical induction that, for z ≠ 1 and all positive integers n, 1 + z + z² + ... + zⁿ⁻¹ = (zⁿ - 1)/(z - 1)[5]
- (ii)By letting z = ½(cos θ + i sin θ), use de Moivre's theorem to deduce that Σ∞ m=1 (½)ᵐ sin mθ = (2 sin θ)/(5 - 4 cos θ)[5]
Mark scheme
Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.
- [1] Shows base case.
- [1] States inductive hypothesis.
- [2] Combines fractions.
- [1] States conclusion.
- [1] States |z|<1 and uses formula for sum to infinity of geometric progression.
- [2] Uses de Moivre's theorem.
- [1] Multiply numerator and denominator by conjugate.
- [1] States imaginary part, AG.
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