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VMO 2018 P1

Source: Vietnam MO 2018 1st day 1st problem

January 11, 2018
calculuslimitSequence

Problem Statement

The sequence (xn)(x_n) is defined as follows: x1=2, xn+1=xn+8−xn+3x_1=2,\, x_{n+1}=\sqrt{x_n+8}-\sqrt{x_n+3} for all n≥1n\geq 1. a. Prove that (xn)(x_n) has a finite limit and find that limit. b. For every n≥1n\geq 1, prove that n≤x1+x2+⋯+xn≤n+1.n\leq x_1+x_2+\dots +x_n\leq n+1.
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