Question

The least positive integer $n,$ for which $\sqrt[3]{3n+1}-\sqrt[3]{3n}<\frac{1}{12}$ is

• A. $6$
• B. $7$
• C. $8$
• D. $9$

Answer 1

The correct option is C $8$

$\sqrt[3]{3n+1}-\sqrt[3]{3n}<\frac{1}{12}$
$\Rightarrow n^{1/3}({(1+\frac{1}{n})}^{1/3}-1)<\frac{1}{12}$
$\Rightarrow n^{1/3}((1+\frac{1}{3n}-\delta )-1)<\frac{1}{12},\delta >0$
$\Rightarrow n^{1/3}\left(\frac{1-3n\delta }{3n}\right)<\frac{1}{12}$
$\Rightarrow n^{2/3}>4(1-3n\delta )$
$\Rightarrow n>8(1-3n\delta {)}^{3/2}$
$\Rightarrow n_{min}=8$