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Two sequences with powers of 2 related to sqrt(3)

Source: Baltic Way 2020, Problem 3

November 14, 2020
Sequenceinequalitiesinequalities proposed

Problem Statement

A real sequence (an)n=0∞(a_n)_{n=0}^\infty is defined recursively by a0=2a_0 = 2 and the recursion formula an={an−12if an−1<3an−123if an−1≥3. a_{n} = \begin{dcases} a_{n-1}^2 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{a_{n-1}^2}{3} & \text{if $a_{n-1}\geq\sqrt 3$.} \end{dcases} Another real sequence (bn)n=1∞(b_n)_{n=1}^\infty is defined in terms of the first by the formula bn={0if an−1<312nif an−1≥3, b_{n} = \begin{dcases} 0 & \text{if $a_{n-1}<\sqrt3$} \\ \frac{1}{2^{n}} & \text{if $a_{n-1}\geq\sqrt 3$,} \end{dcases} valid for each n≥1n\geq 1. Prove that b1+b2+⋯+b2020<23. b_1 + b_2 + \cdots + b_{2020} < \frac23.
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