How is it that $(2^k+1)3 \gt (2^k+1)2\gt 2^{k+1}+2\gt 2^{k+1}+1$? I am working on a mathematical induction worksheet, and my professor gave us the key. I have run across something that makes zero sense to me, so please explain if you can.
Additional info: $k \ge2$
Here is what she has:
What needs to be shown: $$ 3^{k+1} \gt 1 +2^{k+1} \tag0$$
$$3^{k+1} = 3^k(3) \gt (2^k+1)3 \tag1$$
This is where I am confused..
$$ \gt (2^k+1)2 \tag2$$
$$ \gt 2^{k+1}+2 \tag3$$
$$ \gt 2^{k+1}+1 \tag4$$


$\text{As is it was mentioned in the comments we have:}$
$(2^k+1) \cdot 3 >(2^k+1) \cdot 2 \Rightarrow (2^k+1) \cdot 3>2^k \cdot 2+2 \Rightarrow  (2^k+1) \cdot 3>2^{k+1}\cdot 2>2^{k+1}+1$
$\text{Finally:}$
$(2^k+1) \cdot 3 >2^{k+1}+1.$
$\text{A general formoula would be:}$
$\ (2^k+1) \cdot 3 >(2^k+1) \cdot c \text{ with  c<3}$

