When n=0, 3n=2n=1, but the inequality is false because 30<1+20.
The inequality must be qualified by integer n>0.
When n=1, 3n=3 and 2n=2 so 31=1+21, and the inequality is true.
The base case is when n=1 (not n=0).
3n+1=3.3n and 2n+1=2.2n.
Assume 3n≥2n+1 then 3n+1=3.3n≥3.2n+3.
3.2n=(1+2)2n=2n+2n+1, 3n+1≥2n+2n+1+3.
Since 2n+3>1, 3n+1≥2n+1.