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Thread: Induction Proof of Sum of n positive numbers

  1. #1
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    Question Induction Proof of Sum of n positive numbers

    I am not sure if this is the right place, but it does appear in my calculus book.

    "
    Proove by induction that for all n real, positive numbers a1, a2, a3, ..., an that follow the rule a1a2a3...an = 1, the following expession is true:

    $\displaystyle \sum_{i=1}^{n} a_{i} \geq n $

    "

    I have tried working it out but I couldn't proove the induction step.

    I would very appreciate any help with this.
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  2. #2
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    Re: Induction Proof of Sum of n positive numbers

    assuming

    $$x y=1 \Longrightarrow x+y\geq 2 $$

    prove that

    $$a b c =1 \Longrightarrow a+b+c \geq 3$$

    if we suppose $a\geq b\geq c$ then $a\geq 1$ since otherwise $a b c <1$.

    Likewise $c\leq 1$ and so we have the inequality $(a-1)(1-c) \geq 0$

    Applying the induction hypothesis to $x=a c$ and $y= b$

    we get a second inequality $a c+b \geq 2$

    Adding these two inequalities we have

    $$a+b+c \geq 3$$
    Thanks from topsquark
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