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Math Help - Inductive Proof of Inequality

  1. #1
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    Inductive Proof of Inequality

    Prove by induction that 3^n >= 1 + 2^n
    I am trying to prove this by showing 3^(k+1) = 3*3^k >= 1 + 2^(k+1), so I was using an inductive substitution of putting 3(1+2^k) on the left side of the inequality, but it doesn't seem to get me anywhere. I can't think of how else to do this.
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  2. #2
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    Assume 3^n \ge 2^n + 1

    Then 3^{n+1} = 3^n \cdot 3 \ge 2^n \cdot 3 = 2^n \cdot 2 + 2^n \ge 2^n \cdot 2 + 1 = 2^{n+1} + 1

    The last step was true since x^n > 1 \forall x>1, n \ge 0
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  3. #3
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    Can you explain this part to me?

    2^n \cdot 3 = 2^n \cdot 2 + 2^n
    Last edited by veronicak5678; October 13th 2010 at 07:32 PM.
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  4. #4
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    \forall a: a \cdot 3 = a \cdot 2 + a or a \cdot 3 = a + a + a = (a+a) + a = 2a + a. We just apply this for the case a = 2^n
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  5. #5
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    Wow. I must have been really tired last night. I see! Thanks for your answers.
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