MathNet 2024-part-04 — 168 problems (EN–VI) Question 93

Let $p \ge 2$ be an integer number. Prove that the sequence $(x_n)_{n \ge 1}$, defined by $x_1 = a \gt 0$ and the recurrence relation $x_{n+1} = x_n + \lfloor \frac{p}{x_n} \rfloor$, $n \in \mathbb{N}^*$, is convergent. Determine its limit depending on the values of the parameter $a$. We denote by $[x]$ the integer part of the real number $x$.
Emil Vasile