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Math Help - proof by induction help

  1. #1
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    proof by induction help

    I've been reviewing mathematical induction and I don't really get how to prove inequalities by induction. I've been working on this problem for a while..

    2^n\leq(n+1)!

    so far I've proved n=1 for the base, then I assume n=k.
    2^k\leq(k+1)!

    then, I use n=(k+1)..
    2^(k+1)\leq((k+1)+1)!

    this is where i get confused, I did this..
    2\bullet(k+1)!=((k+1)+1)!
    I multiplied by 2 because you can rewrite 2^(k+1) as 2^k\bullet2 but i'm not sure if multiplying by two on the other side is right since it's an inequality and not an equation like what i've been working with up until now. I would really appreciate help with trying to solve this
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  2. #2
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    There's actually no need for induction...

    \displaystyle 2^n = 2\cdot 2 \cdot 2 \cdot \dots \cdot 2 ( \displaystyle n times)


    \displaystyle (n+1)! = (n+1)\cdot n \cdot (n - 1) \cdot \dots \cdot 3 \cdot 2 (again, \displaystyle n terms).


    Since each has \displaystyle n terms and each term in \displaystyle n! is no less than \displaystyle 2, that means

    \displaystyle 2 \cdot 2 \cdot 2 \cdot \dots \cdot 2 < (n + 1) \cdot n \cdot (n - 1) \cdot \dots \cdot 3 \cdot 2

    \displaystyle 2^n < (n+1)! for \displaystyle n > 1 (if \displaystyle n = 1 then \displaystyle 2^n = n!).
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  3. #3
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    the question asks to prove in by mathematical induction, but thanks for the help.
    i don't know how i ended up with two threads. sorry, i didn't mean to post this one, i think i was trying to preview it.
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  4. #4
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    Quote Originally Posted by yaykittyeee View Post
    I've been reviewing mathematical induction and I don't really get how to prove inequalities by induction. I've been working on this problem for a while..

    2^n\leq(n+1)!

    so far I've proved n=1 for the base, then I assume n=k.
    2^k\leq(k+1)!

    then, I use n=(k+1)..
    2^(k+1)\leq((k+1)+1)!

    this is where i get confused, I did this..
    2\bullet(k+1)!=((k+1)+1)!
    I multiplied by 2 because you can rewrite 2^(k+1) as 2^k\bullet2 but i'm not sure if multiplying by two on the other side is right since it's an inequality and not an equation like what i've been working with up until now. I would really appreciate help with trying to solve this

    Well,

    2^{k+1}=2^k\cdot 2<(k+1)!\cdot 2 \,\,(Ind.\,\, hypothesis) <(k+1)!(k+2) \,\,(since \,\,2 < k+2)=(k+2)! . Q.E. D.

    Tonio
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