International Mathematics Competition
for University Students
2024

IMC2024: Day 2, Problem 8

Problem 8. Define the sequence \(\displaystyle x_1,x_2,\ldots\) by the initial terms \(\displaystyle x_1=2\), \(\displaystyle x_2=4\), and the recurrence relation

\(\displaystyle x_{n+2}=3x_{n+1}-2x_n+\frac{2^ n}{x_n}\quad\text{for }n\geq 1.\)

Prove that \(\displaystyle \displaystyle\lim\limits_{n \to \infty}\frac{x_n}{2^ n}\) exists and satisfies

\(\displaystyle \frac{1+\sqrt{3}}{2}\leq\lim\limits_{n \to \infty}\dfrac{x_n}{2^ n}\leq\frac{3}{2}.\)

Karen Keryan, Yerevan State University & American University of Armenia, Armenia

Hint: Prove that \(\displaystyle 2x_n\le x_{n+1}\le 2x_n+n\).

  Solution