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.

\begin{align*}
1+2^{k+1}&=1+2\cdot 2^{k}\\
&\leq 1+2\cdot(3^{k}-1)\\
&=2\cdot 3^{k}-1\\
&\leq 2\cdot 3^{k}\\
&\leq 3\cdot 3^{k}\\
&=3^{k+1}.
\end{align*}
