Can a simple plot be used as a proof-without-words?

Question

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 $n\geq2$.

Answer 1

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: $$\begin{align} 3^{n+1}+1&=3\cdot(3^n+1)-2\\ &\gt3\cdot2^{n+1}-2 &\text{ (by the induction hypothesis)}\\ &=(2\cdot2^{n+1})+(2^{n+1}-2)\\ &\gt 2\cdot 2^{n+1} &\text{ (since $2^{n+1}\gt 2)$}\\ &=2^{(n+1)+1} \end{align}$$

Answer 2

Repeating comments as requested

The chart certainly suggests (rather than proves) there is no equality above $n=2$ as the blue line is higher and seems to be growing faster. There are easy proofs of this.

$2^{1+n}=1+3^n$ has two solutions in the real numbers: the derivatives of the two sides are equal only at $n=\dfrac{\log_e(2)+\log_e(\log_e(2))-\log_e(\log_e(3))}{\log_e(3)-\log_e(2)} \approx 0.5736$ so there is at most one solution below and at most one solution above this. So the only solutions are $n=0$ and $n=1$.