# Thread: Use induction

1. ## Use induction

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?

2. ## Re: Use induction

Originally Posted by 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?
first of all, your last inequality is wrong. why?

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

3. ## Re: Use induction

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