Question 6
For every positive integer $n$ show that
$$
[\sqrt{n} + \sqrt{n+1}] = [\sqrt{4n+1}] = [\sqrt{4n+2}] = [\sqrt{4n+3}]
$$
where $[x]$ is the greatest integer less than or equal to $x$ (for example $[2.3]=2$, $[\pi]=3$, $[5]=5$).
$$
[\sqrt{n} + \sqrt{n+1}] = [\sqrt{4n+1}] = [\sqrt{4n+2}] = [\sqrt{4n+3}]
$$
where $[x]$ is the greatest integer less than or equal to $x$ (for example $[2.3]=2$, $[\pi]=3$, $[5]=5$).