# Simple Proof by Induction

• October 24th 2009, 10:26 PM
Roam
Simple Proof by Induction
Use induction to prove that for every integer $n \geq 4$, $3^n > n^3$.

Here's the solution:

http://img32.imageshack.us/img32/5042/14616641.gif

I don't understand where the term $k^3+3k^2+4k$ came from in the middle line. Could anyone explain?
• October 24th 2009, 11:35 PM
Defunkt
Quote:

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

Here's the solution:

http://img32.imageshack.us/img32/5042/14616641.gif

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, $k^3+k^3+k^3 > k^3+k^3+4k > k^3+3k^2 + 4k \Rightarrow k^3 + k^3 + k^3 > k^3 + 3k^2 + 4k$