Prove that of the two equations

$\displaystyle \left\lfloor\sqrt{n}+\sqrt{n+1}\right\rfloor=\left \lfloor\sqrt{n}+\sqrt{n+2}\right\rfloor$,

$\displaystyle \left\lfloor\sqrt[3]{n}+\sqrt[3]{n+1}\right\rfloor=\left\lfloor\sqrt[3]{n}+\sqrt[3]{n+2}\right\rfloor$

the first holds for every positive integer $\displaystyle n$, but the second does not.

I'm pretty much stuck on this one. Any help would be much appreciated. Thanks!