Question

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

• A. $k\le \left(2\right[a]+1{)}^2$
• B. $k\le \left(\right[a]+1{)}^4$
• C. $k\le 2^{\left[a\right]+1}$
• D. $k\le \frac{1}{a-\left[a\right]}+1$

Answer 1

The correct option is D $k\le \frac{1}{a-\left[a\right]}+1$

Suppose $\left[a\right]=I+f$ where $I$ is integer part and $f$ is fractional so if $f$ tends to zero, $k$ becomes infinite as $\left[a^k\right]=[a{]}^k$.

If $k$ tends to $0$ now check the options to get D as in D option if $f$ tends to 0 $k$ tends to infinity.