Use strong induction to prove that $nleq3^{n/3}$ for every integer $ngeq0$

Use strong induction to prove that $n\leq3^{n/3}$ for every integer $n\geq0$

According to steps of Strong Induction,
1) I assume the predicate as $P(n)$:
$n\leq3^{n/3}$ for every integer $n\geq0$

2) Check the base case $P(0)$ is true, then apply induction on $n$, assume that $P(n)$ holds for integers from $0$ to $n$.

3) Then I need to prove $P(n+1)$ also holds to prove the predicate holds for all integer $n\geq0$.

But I am in trouble on step 3, I found that the induction on $n$ doesn't have any help to prove $P(n+1)$. I think there must be another helpful induction on $n$, but I can't figure it out. Please help, thanks.

Answer

Start by proving $P\left(0\right),P\left(1\right),P\left(2\right),P\left(3\right)$
and $P\left(4\right)$.

Then for $n\geq4$ we have by strong induction: $$3^{\frac{n+1}{3}}=3.3^{\frac{n-2}{3}}\geq3\left(n-2\right)=n+1+\left(2n-7\right)\geq n+1$$