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Proof by mathematical induction — Question 15

QCAA 2023, Paper 1 · 5 marks

Q15 · 2023 · Technology-freeSimple familiar5 marks

QUESTION 15 (5 marks)

The sum of a geometric progression with nn terms, where the first term is 1 and the common ratio is rr, is given by 1+r+r2+r3+⋯+rn−1=rn−1r−1(r≠1).1+r+r^2+r^3+\cdots+r^{n-1}=\frac{r^n-1}{r-1}\quad(r\ne1). Prove that this rule is true ∀n∈Z+\forall n\in\mathbb Z^+ using mathematical induction by completing the steps of the proof as indicated.
a)
Initial statement:
[1 mark]
b)
Assuming the rule is true for n=kn=k, 1+r+r2+r3+⋯+rk−1=rk−1r−1(r≠1).1+r+r^2+r^3+\cdots+r^{k-1}=\frac{r^k-1}{r-1}\quad(r\ne1). Inductive step:
[3 marks]
c)
Conclusion:
[1 mark]
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