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 $8$

$\sqrt[3]{3n+1}-\sqrt[3]{3n}<\frac{1}{12}$
${(n+1)}^{1/3}<n^{1/3}+\frac{1}{12}$
Cube Both side are get
$\left(n^{1/3}\right)(n^{1/3}+\frac{1}{12})>\frac{1727}{432}$
So, $n=8$ only possible least positive integer