<s>[INST] Use the PMI to prove the following for all natural numbers $3^n≥1+2^n$ Use the PMI to prove the following for all natural numbers
$3^n≥1+2^n$
Base Case: $n=1$
$3^1≥1+2^1$ 
$3 ≥ 3$, which is true
Inductive Case:
Assume $3^k ≥ 1+2^k$
[Need to Show for k+1]
$3^{(k+1)} \ge 1+2^{(k+1)}$
Now from here i always get stuck trying to show this next step. I already saw that this same question is asked on this website but i still don't understand why the steps are correct.
for example,
$3^{(k+1)}=3^k⋅3 \ge (1+2^k)3$
how does the RHS turn into that? is there an algebra step that i just dont remember learning?
 [/INST] Assume $$3^k\ge 1+2^k$$
Add $2^k$ to both sides:
$$3^k+2^k\ge 1+2^k+2^k=1+2^{k+1}$$
Now notice that $3^{k+1}=3\cdot 3^k=3^k+2\cdot 3^k\ge 3^k+2^k$
So, we combine $3^k+2^k\ge 1+2^{k+1}$ with $3^{k+1}\ge 3^k+2^k$ and get 
$$3^{k+1}\ge 1+2^{k+1}$$
</s>