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(a_n) and (b_n) have finite limits

Source: 2023 VMO D1P1

August 18, 2023
algebra

Problem Statement

Consider the sequence (an)(a_n) satisfying a1=12,an+1=3an+1−an3a_1=\dfrac{1}{2},a_{n+1}=\sqrt[3]{3a_{n+1}-a_n} and 0≤an≤1,∀n≥1.0\le a_n\le 1,\forall n\ge 1.
a. Prove that the sequence (an)(a_n) is determined uniquely and has finite limit.
b. Let bn=(1+2.a1)(1+22a2)...(1+2nan),∀n≥1.b_n=(1+2.a_1)(1+2^2a_2)...(1+2^na_n), \forall n\ge 1.
Prove that the sequence (bn)(b_n) has finite limit.
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