Proof by Induction that $3^n ≥ 1+2^n$

induction inequality

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!


Solution 1:

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

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