International Mathematics Competition
for University Students
2018

IMC2018: Day 2, Problem 7

Problem 7. Let \(\displaystyle (a_n)_{n=0}^\infty\) be a sequence of real numbers such that \(\displaystyle a_0=0\) and

\(\displaystyle a_{n+1}^3=a_n^2-8 \quad \text{for} \quad n=0,1,2,\ldots \)

Prove that the following series is convergent:

(Proposed by Orif Ibrogimov, National University of Uzbekistan)

Hint: Find a constant \(\displaystyle 0<q<1\) such that \(\displaystyle |a_{n+2}-a_{n+1}|<q|a_{n+1}-a_{n}|\).

  Solution