<s>[INST] Solve by Mathematical Induction: $2^n+1 \leq 3^n$ I'm very confused on how to prove this.  
By using Mathematical Induction prove that, for all positive integers $n$ the
following inequality holds: 
$$2^n + 1 ≤ 3^n$$
 [/INST] For $n = 1$,
$$3^1 = 3 \ge 2^1 + 1$$
Assume that $2^k + 1 \le 3^k$ for some $k\in \mathbb N$.
For $n = k+1$,
$$\begin{align*}
3^{k+1} &= 3\cdot 3^k\\
&\ge 3\cdot(2^k+1)\\
&= 3\cdot 2^k + 3\\
&= 2\cdot2^k + 1 + 2^k + 2\\
&\ge 2^{k+1} + 1
\end{align*}$$
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