Question

Prove that $2^n>n^2$if n is an integer greater than 4.

Short answer

P(n) is true for all positive integers n>4.

Step-by-step solution

Step: 1

Put n=5,

$$\begin{gathered} 2^5>5^2 \\ 32<32 \end{gathered}$$

It is true for n=5.

Step: 2

Let P(k) be true where k is greater than 5.

$2^k>k^2$

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

Step: 3

$\begin{array}{r}{2}^{k+1}={2.2}^k\\ =2^k+2^k\\ >k^2+k^2\\ =k^2+k\cdot k\\ >k^2+4k\\ =(k+1{)}^2\end{array}$

It is true for p(k+1).