1. ## Use induction

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

My attempt:

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

Assume that the statement is true for n.

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

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

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

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

What's the easiest way of proving this?

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

2. ## Re: Use induction

Originally Posted by alexmahone
Use induction to prove that $3^n>n^4$ if $n\geq8$.

My attempt:

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

Assume that the statement is true for n.

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

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

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

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

What's the easiest way of proving this?

Edit: The above inequality is obviously not true for large n. What should I do, instead?
first of all, your last inequality is wrong. why?

also, you shouldn't have expanded $(n+1)^4.$ just take the fourth root to get the inequality $\sqrt[4]{3} > 1 + \frac{1}{n},$ which is true for all $n \geq 4$ because $1 + \frac{1}{n} \leq 1+ \frac{1}{4} = 1.25$ but $\sqrt[4]{3} > 1.3$.

3. ## Re: Use induction

Originally Posted by NonCommAlg
first of all, your last inequality is wrong. why?
Oops, it should have been $-2n^4+4n^3+6n^2+4n+1<0$.