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Math Help - i struggle with proofs by induction

  1. #1
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    i struggle with proofs by induction

    Prove by induction that for all integers n greater than or equal to 2, 2^n+1 < 3^n
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  2. #2
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    Quote Originally Posted by leinadwerdna View Post
    Prove by induction that for all integers n greater than or equal to 2, 2^n+1 < 3^n
    This is how I would do it:

    For n=2, 2^3 < 3^2, 8<9, true for n=2

    Assume for n=k, this gives you 2^{k+1} < 3^k

    Now the induction step, prove for n=k+1

    2^{k+2} < 3^{k+1}

    2 \cdot 2^{k+1} < 3 \cdot 3^{k}

    Dividing both sides by 2

    2^{k+1} < \frac{3}{2} \cdot 3^{k}

    Therefore if 2^{k+1} < 3^k, then 2^{k+1} is definitely < \frac{3}{2} \cdot 3^{k}, true for all n=k+1 if true for n=k

    True for n=2, therefore true for all integers greater than or equal to 2.

    QED
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  3. #3
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    for the induction step couldnt you also do with 2^(k+1)<3^k multiply both sides by 2 of that so you get 4(2^k) < 2(3^k). And since we are trying to prove 4(2^k)<3(3^k) this obviously proves it since 2 is less than 3

    does this make any sense
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