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Iterated fractions

Source: Kvant Magazine No. 11-12 2022 M2726

March 7, 2023

Kvantalgebra

Problem Statement

Let

a_1=1

and

a_{n+1}=2/(2+a_n)

for all

n\geqslant 1

. Similarly,

b_1=1

and

b_{n+1}=3/(3+b_n)

for all

n\geqslant 1

. Which is greater between

a_{2022}

and

b_{2022}

?
Proposed by P. Kozhevnikov