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a_{n+2}/ a_n = 1/ 4 , a_{k+1}/a_k+ a_{n+1}/a_n}=1 , geometric progression

Source: Greece NMO 1996 p1

May 26, 2019
recurrence relationalgebrageometric progressionSequencegeometric sequence

Problem Statement

Let ana_n be a sequence of positive numbers such that: i) an+2an=14\dfrac{a_{n+2}}{a_n}=\dfrac{1}{4}, for every n∈N⋆n\in\mathbb{N}^{\star} ii) ak+1ak+an+1an=1\dfrac{a_{k+1}}{a_k}+\dfrac{a_{n+1}}{a_n}=1, for every k,n∈N⋆ k,n\in\mathbb{N}^{\star} with ∣k−n∣≠1|k-n|\neq 1. (a) Prove that (an)(a_n) is a geometric progression. (n) Prove that exists t>0t>0, such that an+1≤12an+t\sqrt{a_{n+1}}\leq \dfrac{1}{2}a_n+t
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