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Prove that $2^n > n$ for all positive integers $n$.

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Let $P(n) : 2^n > n$
When $n=1$, $2^1 > 1$. Hence $P(1)$ is true.
Assume that $P(k)$ is true for some positive integers $k$, i.e.,
$ \qquad 2^k >k \qquad$ ----------(1)
We shall now prove that $P(k+1)$ is true whenever $P(k)$ is true.
Multiplying both sides of (1) by 2, we get
$\qquad2.2^k > 2k$
i.e., $\: \: 2^{k+1} > 2k=k+k>k+1$
Therefore $P(k+1)$ is true when $P(k)$ is true. Hence by principle of mathematical induction, $P(n)$ is true for every positive integer $n$
answered May 2, 2014 by thanvigandhi_1
edited May 2, 2014 by thanvigandhi_1

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