Proving Inequality

**UC151CPR** · Oct 15th 2009, 08:14 PM

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

**Grandad** · Oct 16th 2009, 02:51 AM

Hello UC151CPR

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 $n \ge1$

Grandad

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