<s>[INST] Proof by Induction that $3^n ≥ 1+2^n$ 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!
 [/INST] 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} $. 
$$ 3^{n+1} = 3^n \cdot 3 \geq(1 + 2^n)3 = 3 + 2^n \cdot 3 = 1 + 2 + 2^n \cdot 3 = 1 + 2 + (1+2)2^n = 1 + 2 + 2^{n+1} + 2 > 1 + 2^{n+1}$$
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