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two sequences

Source: Vietnam Mathematical Olympiad 2016, Day 1, Problem 2

January 6, 2016
algebralinear combination

Problem Statement

a) Let (an)(a_n) be the sequence defined by an=ln⁡(2n2+1)−ln⁡(n2+n+1)  ∀n≥1.a_n=\ln (2n^2+1)-\ln (n^2+n+1)\,\,\forall n\geq 1. Prove that the set {n∈N∣ {an}<12}\{n\in\mathbb{N}|\,\{a_n\}<\dfrac{1}{2}\} is a finite set; b) Let (bn)(b_n) be the sequence defined by an=ln⁡(2n2+1)+ln⁡(n2+n+1)  ∀n≥1a_n=\ln (2n^2+1)+\ln (n^2+n+1)\,\,\forall n\geq 1. Prove that the set {n∈N∣ {bn}<12016}\{n\in\mathbb{N}|\,\{b_n\}<\dfrac{1}{2016}\} is an infinite set.
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