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?