For a real number $r$ let $[r]$ denote the largest integer less than or equal to r. Let $a>1$ be a real number which is not an integer, and let $\mathrm{k}$ be the smallest positive integer such that $\left[\mathrm{a}^{\mathrm{k}}\right]>[\mathrm{a}]^{\mathrm{k}}$. then which of the following statements is always true?

• A

  $k \leq 2([a]+1)^{2}$
• B

  $k \leq([a]+1)^{4}$
• C

  $k \leq 2^{[a]+1}$
• Correct answer
  D

  $k \leq \frac{1}{a-[a]}+1$

Hence, the correct answer is option: 'D'.