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8

Part of 2024 IMC

Problems(1)

An almost-linear recurrence featuring 2^n

Source: IMC 2024, Problem 8

8/8/2024
Define the sequence x1,x2,…x_1,x_2,\dots by the initial terms x1=2,x2=4x_1=2, x_2=4, and the recurrence relation x_{n+2}=3x_{n+1}-2x_n+\frac{2^n}{x_n}   \text{for}   n \ge 1. Prove that lim⁡n→∞xn2n\lim_{n \to \infty} \frac{x_n}{2^n} exists and satisfies 1+32≤lim⁡n→∞xn2n≤32.\frac{1+\sqrt{3}}{2} \le \lim_{n \to \infty} \frac{x_n}{2^n} \le \frac{3}{2}.
recurrence relationSequenceslimit