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Thread: Proving a formula

  1. #1
    Member mohammadfawaz's Avatar
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    Proving a formula

    Hello all,

    I need help proving the following:
    if a,b,c,... are n real positive number such that a+b+c+... = 1, then
    a^2 + b^2 + c^2 + ... >=1/n

    Thank you
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  2. #2
    Super Member Bacterius's Avatar
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    Hello,
    denote all these numbers $\displaystyle a_1$, $\displaystyle a_2$, ... $\displaystyle a_n$, with the property :

    $\displaystyle \sum_{k = 1}^{n} a_k = 1$

    Now note that when $\displaystyle a_k = \frac{1}{n}$, we have $\displaystyle a_k^2 = \frac{1}{n^2}$, and it follows that :

    $\displaystyle \sum_{k = 1}^{n} a_k^2 = \sum_{k = 1}^{n} \frac{1}{n^2} = n \times \frac{1}{n^2} = \frac{1}{n}$

    This is not a proof in itself : it is there to help you grasp what this problem means. All you have to prove is that there is no solution for $\displaystyle a_k$ where $\displaystyle \sum_{k = 1}^{n} a_k^2 < \frac{1}{n}$.

    Here is an idea : say $\displaystyle a_1 = \frac{1}{x}$. Therefore, the following holds : $\displaystyle 0 < a_2 < 1 - \frac{1}{x}$, and so on following the same logic with $\displaystyle a_3$, $\displaystyle a_4$, .... Take the squares of these inequalities, and see where you get You will very likely stumble upon a difficulty, that you will probably be able to express in mathematical terms and that will lead to the solution to your problem
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  3. #3
    Member mohammadfawaz's Avatar
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    Hey..
    Thank you for the reply...
    I did what we said: I bounded each number between 0 and $\displaystyle 1-\frac{1}{x}$. When added them all, I got: $\displaystyle \frac{1}{x^2}<\sum _{k=1}^n a_k^2 < \frac{n}{x^2} - \frac{2(n-1)}{x} + n - 1$. But i didn't know what to do next. How can I prove from here that there is no solution for the inequality $\displaystyle \sum_{k = 1}^{n} a_k^2 < \frac{1}{n}$?

    Thank you
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  4. #4
    Super Member Bacterius's Avatar
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    EDIT : forget this stuff, I got it wrong. Hang on, I'm thinking of something ...

    Yeah, I think I got it. Say you got your numbers, all equal to each other (so they all equal $\displaystyle \frac{1}{n}$). If you add some value $\displaystyle x$ to one number, you need to report the opposite of this value onto the other numbers to balance the equality. Now which values of $\displaystyle x$ don't allow any balancing of the other numbers that satisfies the sum of the squares being less than $\displaystyle \frac{1}{n}$ ? Try to express this in mathematical terms. You should find that $\displaystyle x$ has no solution ... your problem follows. I might post a proof, I'm actually as stumped as you on this one
    Last edited by Bacterius; Feb 28th 2010 at 03:00 AM.
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  5. #5
    Member u2_wa's Avatar
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    Quote Originally Posted by mohammadfawaz View Post
    Hello all,

    I need help proving the following:
    if a,b,c,... are n real positive number such that a+b+c+... = 1, then
    a^2 + b^2 + c^2 + ... >=1/n

    Thank you
    Hello mohammadfawaz
    If $\displaystyle a+b+c=1$
    Let A=average of $\displaystyle a+b+c$
    we can say that $\displaystyle a^2+b^2+c^2>=A^2+A^2+A^2$
    Similarly,
    $\displaystyle a+b+c+d+.....=1, A=\frac{1}{n}
    $

    $\displaystyle a^2+b^2+c^2+....>=nA^2$
    $\displaystyle a^2+b^2+c^2+....>=\frac{1}{n}$
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  6. #6
    Member mohammadfawaz's Avatar
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    hello u2_wa

    There is one small thing I didn't get in your proof:
    by saying $\displaystyle a^2+b^2+c^2>=A^2+A^2+A^2$, you're assuming that: $\displaystyle a^2>=A^2$ and $\displaystyle b^2>=A^2$ and $\displaystyle c^2>=A^2$. But if A is the average of a,b and c, then A may be smaller or larger than a, b or c. Would you please explain how did you get the inequality?

    Thank you
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  7. #7
    Member u2_wa's Avatar
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    Quote Originally Posted by mohammadfawaz View Post
    hello u2_wa

    There is one small thing I didn't get in your proof:
    by saying $\displaystyle a^2+b^2+c^2>=A^2+A^2+A^2$, you're assuming that: $\displaystyle a^2>=A^2$ and $\displaystyle b^2>=A^2$ and $\displaystyle c^2>=A^2$. But if A is the average of a,b and c, then A may be smaller or larger than a, b or c. Would you please explain how did you get the inequality?

    Thank you
    Hello mohammadfawaz:

    Certainly I am not assuming $\displaystyle a^2>=A^2$ and $\displaystyle b^2>=A^2$ and $\displaystyle c^2>=A^2$

    I am saying that sum of $\displaystyle a^2+b^2+c^2$ will always be equal to or greater than
    $\displaystyle A^2+A^2+A^2$

    To prove $\displaystyle a^2+b^2+c^2>=A^2+A^2+A^2$

    $\displaystyle a^2+b^2+c^2>=3A^2$

    $\displaystyle a^2+b^2+c^2>=3(\frac{a+b+c}{3})^2$

    $\displaystyle \frac{3a^2+3b^2+3c^2-(a+b+c)^2}{3}>=0$

    $\displaystyle \frac{(a-c)^2+(a-b)^2+(b-c)^2}{3}>=0$ (PROOF)

    Same rule applies to $\displaystyle a+b+c+d+.....=nA$

    Hope this helps!
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