Question

Prove that$3^n<n!$ if n is an integer greater than 6.

Short answer

P(n) is true for all positive integers n>6.

Step-by-step solution

Step: 1

Put n=7,

$$\begin{gathered} 3^7<7! \\ 2128<5040 \end{gathered}$$

It is true for n=7.

Step: 2

Let P(k) be true where k is greater than 6.

$3^k<k!$

We need to prove it is true for p(k+1).

Step: 3

$\begin{array}{r}{3}^{k+1}={3.3}^k\\ <3.k!\\ <(k+1)\cdot k!\\ =(k+1)!\end{array}$

It is true for p(k+1).