Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.
Source: 2011 Saudi Arabia Pre-TST February 2.1
Tags: number theory, Integer
Let $n$ be a positive integer. Prove that the interval $I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right)$ does not contain any integer.