Chapter A2, Problem 7E

To determine

To show: $2^n>n^2$ if $n$ is large enough.

Explanation of Solution

Proof:

Step 1: The inequality holds for $n=5$ because $2^5=32>25=5^2$

Step 2: Suppose $2^k>k^2$

For $k\ge 3,k>2+\frac{1}{k}$ and so, $k^2>2k+1$ . Hence $2^k>2k+1$

So, $2^{k+1}=2^k+2^k>k^2+2k+1={\left(k+1\right)}^2$

Hence, by principle of mathematical induction, this is true for all $n\ge 5$ .

Conclusion:

By principle of mathematical induction, this is true for all $n\ge 5$