Q:

Prove inequality $n<3^n$ using mathematical induction

Prove that $n<3^n$ where $n \in \mathbb N$,
when $ n=1 $, I have proved it's true.
And assumed when $n = p$ , $p<3^p$ is true.
Can any body help me in showing that it is true for $n =p+1$

A:

We have as our inductive hypothesis (IH) that $\,p \lt 3^p$.
So we need to show $$p+1 \lt 3^{p+1} = 3 \cdot 3^p.$$
Can you see that $$p + 1 \quad \overset{IH}{\lt}\quad 3^p + 1 \quad \lt \quad 3\cdot 3^p \;= \;3^{p+1}, \quad \forall p \in \mathbb N$$

