Question

Prove by induction that 3^n is greater than or equal to n^2 for all positive integers greater than or equal to 1

Answer 1

Let n=1

31>12

Let n=2

32>22

Assume P holds for n=k

3k>k2

Let n=k+1

3k+1>(k+1)23×3k>(k+1)2

From here I can not find where to go to finish the proof.

The main part of the question is the proof, however; I would like to also know if using n=k+1 is always the way to go? I have only done a few proofs by induction and so far to my understanding is that the whole point is to prove the function, series or statement for all positive integers. Is there a special way to go about these types of problems when only given an inequality or a single statement? In comparison to being give a sequence and told what that sequence as a function is, i.e 1+2+3+...+n=n(n+1)2 as an example as an easy sequence to prove with n=k+1