TITLE: Can a simple plot be used as a proof-without-words?
QUESTION [0 upvotes]: Can this simple plot be used as a proof-without-words?
Edit "No, it suggests but does not prove."
Plot of $2^{1 + n} = 1 + 3^n:$

Motivated by this question, I reworked the non-loopback inequalities for Collatz and Waring into an equality:
$2^{1 + n} = 1 + 3^n$. It's easy to see the equality fails when $n\geq2$. 
Edit $2^{1 + n} \neq 1 + 3^n$ is solvable for non-negative integer $n$, and is $n\geq2$.

REPLY [2 votes]: Here's an easy inductive calculus-free proof for $n\geq 2$.  Start by showing that $3^{2}+1=10\gt2^{2+1}=8$.  Then:
$$\begin{align}
3^{n+1}+1&=3\cdot(3^n+1)-2\\
&\gt3\cdot2^{n+1}-2 &\text{ (by the induction hypothesis)}\\
&=(2\cdot2^{n+1})+(2^{n+1}-2)\\
&\gt 2\cdot 2^{n+1} &\text{ (since $2^{n+1}\gt 2)$}\\
&=2^{(n+1)+1}
\end{align}$$
{"set_name": "stack_exchange", "score": 0, "question_id": 1204058}