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#### sweatingbear

##### Member

- May 3, 2013

- 91

$3^4 = 81 > 4^3 = 64 $

Assume the inequality holds for $n = p $ i.e. $3^p > p^3$ for $p \geqslant 4$. Then

$3^{p+1} > 3p^3$

$p \geqslant 4$ implies $3p^3 \geq 192$, as well as $(p+1)^3 \geqslant 125$. Thus $3p^3 > (p+1)^3$ for $p \geqslant 4$ and we have

$3^{p+1} > 3p^3 > (p+1)^3$

which concludes the proof.

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