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On the relation between some arithmetic and geometric progressions

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July 7, 2020
arithmetic sequencegeometric sequencebijectionpermutation of the naturalsalgebra

Problem Statement

Let (an)n≥1 \left( a_n \right)_{n\ge 1} be a non-constant arithmetic progression of positive numbers and (gn)n≥1 \left( g_n \right)_{n\ge 1} be a non-constant geometric progression of positive numbers satisfying a1=g1 a_1=g_1 and a2019=g2019. a_{2019} =g_{2019} . Specify the set {k∈N∣ak≤gk} \left\{ k\in\mathbb{N} \big| a_k\le g_k \right\} and prove that it bijects the natural numbers.
Gheorghe Rotariu
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