Thread: Proving Inequality

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

Hello UC151CPR

Originally Posted by UC151CPR

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

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

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

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

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

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

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

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

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

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

Grandad