True/False.

Algebra Level 3

Let $a >1$ be a real number which is not an integer, and let $k$ be the smallest positive integer such that $\lfloor a^k \rfloor >\lfloor a \rfloor ^k$ , then which of the following statement is always true?

Details:

$\lfloor r\rfloor$ denotes the largest integer less than or equal to $r$ .
$\{r\}$ denotes the fractional part of $r$ .

k\leq\frac{1+\{a\}}{\{a\}}

k\leq2(\lfloor(a)\rfloor+1)^2

k\leq2^{\lfloor(a)\rfloor+1}

k\leq(\lfloor(a)\rfloor+1)^4

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True/False.

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