Proving Inequalities With Mathematical Induction I'm currently working on this problem:
$$ 1 + 2^n + ≤ 3^n \text{ for all }  ≥ 1 $$
So far, I have:
Basis Step:
$ 1 + 2^1 ≤3^1 $
$ P(1) \text{ is true} $
Inductive Step: Assume P(k) holds, prove P(k+1) 
$P(k) = 1 + 2^k ≤ 3k$
$ P(k+1) = 1 + 2^{k+1} ≤ 3^{k+1} \text{ (I.H.)}$
$ 1 + 2^{k+1} = 1 + 2 * 2^k$
$ \quad \quad \, \, \quad  = 2 * 2^k + 1 ≤ 2 * 3^k$
But now, I'm unsure what to do next. Any help would be aprreciated!
Thank you.

The inductive step:
$$1+2^{n+1}=1+2\cdot 2^n=1+2^n+2^n\le 3^n+2^n\le 3^n+3^n+3^n=3\cdot 3^n=3^{n+1}.$$
Without induction:
$$1+2^n\le 3^n=(1+2)^n=1+A+2^n, \ \ A\ge 0.$$
