## [ Algebraic inequalities (other).] Induction (other). $\quad]$

For which natural numbers $n$ does the inequality $2^{n} \geq n^{3}$ hold?

#
$2^{9}=526n^{3}$ for $n>9$. The base case ($n=10$) is obvious.

Induction step. $2^{n+1}=2 \cdot 2^{n}>2 n^{3}>(n+1)^{3}$. Indeed, $\frac{(n+1)^{3}}{n^{3}}=\left(1+\frac{1}{n}\right)^{3}=1+\frac{3}{n}+\frac{3}{n^{2}}+\frac{1}{n^{3}}<2$ for $n>9$.

## Answer

For $n \geq 10$.