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IMC 2010 - Problem 3

Source:

July 26, 2010
limitfunctionreal analysisreal analysis unsolved

Problem Statement

Define the sequence x1,x2,...x_1, x_2, ... inductively by x1=5x_1 = \sqrt{5} and xn+1=xn2−2x_{n+1} = x_n^2 - 2 for each n≥1n \geq 1. Compute lim⁡n→∞x1⋅x2⋅x3⋅...⋅xnxn+1\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}.
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