Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Proof of the nontrivial expression

  1. Apr 13, 2014 #1
    I'm looking for a proof of a validity of the inequation:
    [itex](n-1)\sum_{i=1}^{n}x_{i}^{2}\neq 2\sum_{i=1,j=1,j<i}^{n}x_{i}x_{j}[/itex]

    Assumptions:
    [itex]n\geq 2[/itex]

    [itex]\exists (i,j),x_{i}\neq x_{j}[/itex]
    [itex]i=1,...,n[/itex]
    [itex]j=1,...,n[/itex]

    I have no idea how to prove those non-trivial expressions.
     
  2. jcsd
  3. Apr 13, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Start by playing around with the expression ... i.e. try it for n=2 and n=3 and see if you can spot a pattern.

    There are lots of approaches to proofs that you know.
    i.e. induction, assume the converse, and so on.
    Have you tried any?

    It is very common that there is no obvious way to proceed but you have to have a go anyway.
     
  4. Apr 13, 2014 #3
    I started expanding the expression for (n=2,3,4), but it's getting a little chaotic ... I haven't found any pattern so far.
    When I try the induction for (n=2) higher, I receive another expression possibly hiding some pattern.
     
  5. Apr 13, 2014 #4
    [itex]n=2[/itex]:
    [itex]x_{1}^{2}+x_{2}^{2}\neq 2x_{1}x_{2}[/itex]


    [itex]n=3[/itex]:
    [itex]2x_{1}^{2}+2x_{2}^{2}+2x_{3}^{2}\neq 2x_{1}x_{2}+2x_{1}x_{3}+2x_{2}x_{3}[/itex]


    [itex]n=4[/itex]:
    [itex]3x_{1}^{2}+3x_{2}^{2}+3x_{3}^{2}+3x_{4}^{2}\neq 2x_{1}x_{2}+2x_{1}x_{3}+2x_{1}x_{4}+2x_{2}x_{3}+2x_{2}x_{4}+2x_{3}x_{4}[/itex]
     
  6. Apr 13, 2014 #5
    And the in-equation IS VALID - it is the proof of ONE stationary point existence for the function of two variables (for entered assumptions) obtained by the method of least squares. It is easy to imagine that the overall difference of measured values and line values is always minimal for some line directive. I just want to learn to prove it mathematically.
     
    Last edited: Apr 13, 2014
  7. Apr 13, 2014 #6

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    So try to prove the converse. Turn the inequality into an equality - then find the condition that must exist t make the equality work. Since you know the statement is true, the condition will contradict at least one of the assumptions.
     
  8. Apr 13, 2014 #7
    Solved out. It wasn't so difficult to find the pattern:
    [itex]\sum_{i=1,j=1,j<i}^{n}(x_{i}-x_{j})^{2}\neq0[/itex]
    That is always valid for the given assumptions.

    Thank you for your help :)
     
  9. Apr 13, 2014 #8

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Yeah, that what it looks like to me -
    Well done :)
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proof of the nontrivial expression
  1. A proof (Replies: 1)

Loading...