Q29 / 60 · KVPY 2015 (SA) · Mathematics

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

  • A
    6
  • B
    7
  • Correct answer
    C
    8
  • D
    9

Solution

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

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

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