# Math Help - Proving Inequality

1. ## Proving Inequality

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

2. 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 $n \ge1$