# Math Help - Inductive Proof of Inequality

1. ## Inductive Proof of Inequality

Prove by induction that 3^n >= 1 + 2^n
I am trying to prove this by showing 3^(k+1) = 3*3^k >= 1 + 2^(k+1), so I was using an inductive substitution of putting 3(1+2^k) on the left side of the inequality, but it doesn't seem to get me anywhere. I can't think of how else to do this.

2. Assume $3^n \ge 2^n + 1$

Then $3^{n+1} = 3^n \cdot 3 \ge 2^n \cdot 3 = 2^n \cdot 2 + 2^n \ge 2^n \cdot 2 + 1 = 2^{n+1} + 1$

The last step was true since $x^n > 1 \forall x>1, n \ge 0$

3. Can you explain this part to me?

$2^n \cdot 3 = 2^n \cdot 2 + 2^n$

4. $\forall a: a \cdot 3 = a \cdot 2 + a$ or $a \cdot 3 = a + a + a = (a+a) + a = 2a + a$. We just apply this for the case $a = 2^n$

5. Wow. I must have been really tired last night. I see! Thanks for your answers.