Prove that $2^{n} > n$ for all positive integers $n$

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Solution

Let $P (n) : 2^{n} > n$
When $n = 1, 2^{1} > 1$ .Hence $P (1)$ is true.
Assume that $P (k)$ is true for any positive integer $k$ ,i.e.,
$2^{k} > k$
we shall now prove that $P (k + 1)$ is true whenever $P (k)$ is true.
Multiplying both sides of $(1)$ by $2$ , we get
${2.2}^{k} > 2 k$
i.e., $2^{k + 1} > 2 k$
$k + k > k + 1$
$∴ 2^{k + 1} > k + 1$

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