Year 12 Maths Extension 2 (2027)
Proof
Induction – Recursive Formulas & Calculus Results
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Questions
Question 1
183378
A sequence is defined by \(u_1 = 3\) and \(u_{n+1} = u_n + 4\) for \(n \ge 1\). Prove by mathematical induction that \(u_n = 4n - 1\) for all positive integers \(n\).
Base \(n=1\): \(u_1 = 3 = 4(1)-1\). Step: \(u_{k+1} = u_k + 4 = (4k-1) + 4 = 4(k+1)-1\). So \(u_n = 4n-1\) for all \(n \ge 1\).
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