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**alexmahone** Use induction to prove that $\displaystyle 3^n>n^4$ if $\displaystyle n\geq8$.

__My attempt:__

For n=8, the statement becomes 6561>4096, which is true.

Assume that the statement is true for n.

Then, $\displaystyle 3^{n+1}>3n^4$

We need to prove that $\displaystyle 3n^4>(n+1)^4$

$\displaystyle \Leftrightarrow 3n^4>n^4+4n^3+6n^2+4n+1$

$\displaystyle \Leftrightarrow -2n^4+4n^3+6n^2+4n+1>0$ for $\displaystyle n\geq8$.

What's the easiest way of proving this?

Edit: The above inequality is obviously not true for large n. What should I do, instead?