Chapter 9.4, Problem 25E

To determine

To prove: The inequality $n!>2^n,n\ge 4$ for every integer values of $n$ . Using mathematical induction.

Explanation of Solution

Given: $n!>2^n,n\ge 4$

The given inequality, $n!>2^n,n\ge 4$

Let, $P\left(n\right)=n!$

Put $n=4$

LHS: $P\left(4\right)=4!=24$

RHS: $2^4=16$

  $24>16$

  $P\left(n\right)$ is true for $n=4.$

So, $P\left(k\right)$ is true for $n=k$ where, $k\in Z^{+}$ .

  $\therefore k!>2^k\mathrm{...}\left(i\right)$

Now, prove for $P\left(k+1\right)$ .

Put $n=k+1$

For LHS,

  $\begin{array}{c}P\left(k+1\right)=\left(k+1\right)!\\ =\left(k+1\right)k!\left[\text{Using}\left(i\right)\right]\\ =\left(k+1\right)2^k\end{array}$

For RHS,

  $\begin{array}{l}=2^{k+1}\\ =2^k\cdot 2\end{array}$

  $\begin{array}{l}\therefore \left(k+1\right)2^k>2\cdot 2^k\\ \Rightarrow k+1>2\left[\text{True}\right]\end{array}$

Hence, $P\left(k+1\right)$ is true.