Simple Proof by Induction

**Roam** · October 24th 2009, 09:26 PM

Use induction to prove that for every integer $n \geq 4$ , $3^n > n^3$ .

Here's the solution:

I don't understand where the term $k^3+3k^2+4k$ came from in the middle line. Could anyone explain?

**Defunkt** · October 24th 2009, 10:35 PM

Originally Posted by Roam

Use induction to prove that for every integer $n \geq 4$ , $3^n > n^3$ .

Here's the solution:

I don't understand where the term $k^3+3k^2+4k$ came from in the middle line. Could anyone explain?

Since $k \ge 4$ , we know that $k^3 > 3k^2$ and also that $k^3 > 4k$ . Therefore, $k^3+k^3+k^3 > k^3+k^3+4k > k^3+3k^2 + 4k \Rightarrow k^3 + k^3 + k^3 > k^3 + 3k^2 + 4k$

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