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Thread: Short prove by induction

  1. #1
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    Smile Short prove by induction

    I need to prove the following for one of my solutions in order to solve a problem:

    For $\displaystyle n \in\mathbb{N}$ if $\displaystyle n>=3^3$ then $\displaystyle 3^n > 79n^2$

    Thank you in advance.
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  2. #2
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    Have you checked that $\displaystyle 3^{3^{3}}>79.(3^{3})^{2}$?

    Then, let $\displaystyle n \geq 3^{3}$ be an integer, and assume that $\displaystyle 3^{n}>79n^{2}$ (induction hypothesis)

    $\displaystyle 79(n+1)^{2}=79n^{2}+79(2n)+79$

    Is it true that, if $\displaystyle n\geq 3^{3}\ $, then $\displaystyle \ 2n\leq n^{2}$ and $\displaystyle 1\leq n^{2}$ ?

    If it's the case, $\displaystyle 79n^{2}+79(2n)+79 \leq 79n^{2}+79n^{2}+79n^{2}=3(79n^{2})$

    What can we conclude using the induction hypothesis?
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