# Proving Inequality

• October 15th 2009, 08:14 PM
UC151CPR
Proving Inequality
Prove that 1+2n <= 3^n by using mathematical induction.
• October 16th 2009, 02:51 AM
Hello UC151CPR
Quote:

Originally Posted by UC151CPR
Prove that 1+2n <= 3^n by using mathematical induction.

First, note that for $n\ge0, 3^n\ge1$

$\Rightarrow 2\cdot3^n\ge 2$ (1)

Then suppose that $P(n)$ is the propositional function $1+2n\le 3^n$

Then $P(n) \Rightarrow 1+2n + 2 \le 3^n+2$

$\Rightarrow 1 +2(n+1) \le 3^n + 2\cdot3^n$, using (1)

$\Rightarrow 1 +2(n+1) \le 3^n(1+2)=3^{n+1}$

So $P(n) \Rightarrow P(n+1)$

$P(1)$ is $1+2\le3^1$, which is true.

Hence by Induction, $P(n)$ is true for all $n \ge1$