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

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beatrice0866 | Student, College Freshman | eNoter

Posted May 3, 2011 at 12:34 PM via web

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

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted May 3, 2011 at 12:41 PM (Answer #1)

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We can prove that 2^n > n for all positive integers n by induction.

First for n = 1, 2^n = 2 which is greater than 1.

Assume 2^n > n

Now 2^(n + 1) should be greater than (n + 1)

2^(n + 1) = 2*2^n > 2n

2n = n + n > n + 1

=> 2^(n + 1) > n + 1 if 2^n > n

This proves that 2^n > n for all positive integers n.

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