Do there exist naturaln numbers $a, b, n \in \mathbb N$ such that $$\sqrt{n}+\sqrt{n+1}<\sqrt{a}+\sqrt{b}<\sqrt{4n+2}?$$
One can easily see that both $a, b$ can't be perfect squares, because $$(\sqrt{n} + \sqrt{n+1})^2 = 2n +1 + 2 \sqrt{n^2+n} < 4n+2,$$ which after trivial transformations is equivalent to $$4n^2+ 4n < 4n^2 + 4n +1.$$