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I've made a few plots and noticed that $\lfloor

Is it true that for positive $x > 1$ and $n \in \mathbb N,\quad n>=2$ the following holds:

$$(\lfloor x \rfloor + 1)^n >= \lfloor x^n \rfloor $$

If it is, how can it be proven? If it is not, will that at least hold when $n=2$? I am interested in the latter case, actually.

Thank you!

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Are you missing something in the description of what you noticed? – Henry May 16 '11 at 22:28
up vote 4 down vote accepted

$\lfloor x\rfloor +1 \gt x$, so $(\lfloor x\rfloor +1)^n\gt x^n$. On the other hand, $\lfloor x^n\rfloor\leq x^n$.

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Hint: $\lfloor x \rfloor + 1 \ge x$

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