Question

Prove that for each nonnegative integer n,$2^n>n$ .

Short answer

It is proved for each nonnegative integer n,$2^n>n$ .

Step-by-step solution

Consider the given expression for P(n)

Consider the following expression for P(n)as follows:

$P\left(n\right)=2^n$

Here, n is any real number.

Check the expression for n = 1 and n = 2

Substitute 1 for n in the equation $P\left(n\right)=2^n$as follows:

$\begin{array}{rcl}P\left(1\right)& =& 2^1\\ & =& 2\end{array}$

Substitute 2 for n in the equation $P\left(n\right)=2^n$ as follows:

$\begin{array}{rcl}P\left(2\right)& =& 2^2\\ & =& 4\end{array}$

Check the expression for n = k.

Substitute k for n in the equation $P\left(n\right)=2^n$as follows:

$P\left(k\right)=2^k$

Check the expression for n = k + 1. 

Substitute $k+1$ for n in the equation $P\left(n\right)=2^n$ as follows:

$\begin{array}{rcl}P\left(k+1\right)& =& 2^{k+1}\\ & =& 2^k\times 2\end{array}$

Hence, for each nonnegative integer n, $2^n>n$.