Year 12 Maths Extension 2 (2027)
Proof
Induction – Recursive (first-order recurrence)
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Questions
Question 1
183409
A sequence is defined by \(u_1 = 2\) and \(u_n = 2u_{n-1}\) for \(n \ge 2\). Prove by mathematical induction that \(u_n = 2^n\) for all positive integers \(n\).
Base \(n=1\): \(u_1 = 2 = 2^1\). Step: \(u_{k+1} = 2u_k = 2\cdot 2^k = 2^{k+1}\). Holds for all \(n \ge 1\).
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