Proof by Induction that $3^n ≥ 1+2^n$

Question

Use the PMI to prove the following for all natural numbers: $3^n ≥ 1+2^n$. I have already verified the base case but am having trouble doing so with the inductive case. Thanks!

Answer 1

Suppose the result holds for the case $n$ : $3^n \geq 1 + 2^n$. Now, to complete the induction proof, we must show $ 3^{n+1} \geq 1 + 2^{n+1} $.

$$ 3^{n+1} = 3^n \cdot 3 \geq(1 + 2^n)3 = 3 + 2^n \cdot 3 = 1 + 2 + 2^n \cdot 3 = 1 + 2 + (1+2)2^n = 1 + 2 + 2^{n+1} + 2 > 1 + 2^{n+1}$$

Answer 2

$3^{n+1}=3(3^n)>=3(1+2^n)=3+3\cdot 2^n=(1+2\cdot 2^n)+2+2^n>1+2^{n+1}$