Prove by the principle of mathematical induction that $2^{n} > n$ for all $n \in N .$

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Solution

Let $P (n)$ be the statement: $2^{n} > n$

$P (1)$ means $2^{1} > 1$ i.e. $2 > 1,$ which is true

$\Rightarrow P (1)$ is true.

Let $P (m)$ be true
$\Rightarrow 2^{m} > m$

$\Rightarrow {2.2}^{m} > 2. m \Rightarrow 2^{m + 1} > 2 m \geq m + 1$

$\Rightarrow 2^{m + 1} > m + 1$

$\Rightarrow P (m + 1)$ is true.

$∴$ $2^{n} > n$ for all $n \in N$

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