Problem

Source: 2023 VMO D1P1

Tags: algebra



Consider the sequence $(a_n)$ satisfying $a_1=\dfrac{1}{2},a_{n+1}=\sqrt[3]{3a_{n+1}-a_n}$ and $0\le a_n\le 1,\forall n\ge 1.$ a. Prove that the sequence $(a_n)$ is determined uniquely and has finite limit. b. Let $b_n=(1+2.a_1)(1+2^2a_2)...(1+2^na_n), \forall n\ge 1.$ Prove that the sequence $(b_n)$ has finite limit.