Hey there! Sign in to join this conversationNew here? Join for free
    • Thread Starter
    Offline

    1
    ReputationRep:
    \displaystyle \sum_{\text{sym}} \frac{a^2}{b} = \frac{a^2}{b} + \frac{b^2}{a} + \frac{a^2}{c} + \frac{c^2}{a} + \frac{b^2}{c} + \frac{c^2}{b}
    • Wiki Support Team
    Offline

    14
    ReputationRep:
    Wiki Support Team
    What's "sym"? Can you give us a context for this?

    This looks like something I might have written in a "roots of equations" context...
    • Thread Starter
    Offline

    1
    ReputationRep:
    Symmetric sum for a,b,c

    The chapter in the book is symmetric and cyclic expressions
    Offline

    16
    ReputationRep:
    Looks good. See my post from ages back that you already know the location of
    • Thread Starter
    Offline

    1
    ReputationRep:
    I got a question about that.

    How come you had  \displaystyle \sum_{sym} f(a,b,c) = 2(a^2+b^2+c^2), where did the two come from? Wouldn't it be the same as the cyclic sum which is also symmetric
    Offline

    16
    ReputationRep:
    \displaystyle \sum_{\mathbf{sym}} f(a,b,c) = f(a,b,c)+f(a,c,b)+f(b,a,c)+f(b,c  ,a)+f(c,a,b)+f(c,b,a) (Definition)

    f(a,b,c)=a^2b^0c^0

    Therefore f(a,b,c)=f(a,c,b)=a^2
    • Thread Starter
    Offline

    1
    ReputationRep:
    Oh so it means you will get b^2 and c^2 twice as well

    So eg.  f(a,b,c) = a^2b^2c^0

    \left \displaystyle \sum_{sym} a^2b^2 = a^2b^2 + b^2a^2 + a^2c^2 + c^2a^2 + b^2c^2 + c^2b^2 \\= 2(a^2b^2 + a^2c^2 + b^2c^2) right?
    Offline

    16
    ReputationRep:
    Exactly. This distinction is quite important and often forgotten.
    • Thread Starter
    Offline

    1
    ReputationRep:
    Alright thanks
    • Thread Starter
    Offline

    1
    ReputationRep:
    So does that mean that \displaystyle \sum_{sym} f(a,b) = f(a,b) + f(a,b) + f(b,a) + f(b,a) or is it just  \displaystyle \sum_{sym} f(a,b) = f(a,b)+ f(b,a)
    Offline

    16
    ReputationRep:
    The second. It is the sum across all permutations of the set \{ a_1, a_2, \ldots a_n \}

    It could be expressed as

    \displaystyle \sum_{\sigma} f(a_{\sigma (1)}, a_{\sigma (2)}, \ldots, a_{\sigma (n)} ) (if you are happy with that notation)
    • Thread Starter
    Offline

    1
    ReputationRep:
    OK so I got a question that says prove  \displaystyle \sum_{sym} a^3 + \sum_{sym} a = \sum_{sym} a^4 + \sum_{sym} a^3b. I proved it on the assumption that the variables a and b are the only ones involved. Is that correct? And is LHS the same as  \displaystyle (a^3+b^3)(a+b) ?
    Offline

    16
    ReputationRep:
    Perhaps you mean

    \displaystyle \left ( \sum_{\mathbf{sym}} a^3 \right ) \left ( \sum_{\mathbf{sym}} a \right ) = \sum_{\mathbf{sym}} a^4 + \sum_{\mathbf{sym}} a^3 b

    ?
    • Thread Starter
    Offline

    1
    ReputationRep:
    Yeah sorry LHS is a product not a sum
    Offline

    16
    ReputationRep:
    Ok

    \displaystyle \left ( \sum_{\mathbf{sym}} a^3 \right ) \left ( \sum_{\mathbf{sym}} a \right ) = (a^3+b^3)(a+b)

    Unless they give more information on the number of variables?
    • Thread Starter
    Offline

    1
    ReputationRep:
    Thanks again
 
 
 
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • Poll
    Would you like to hibernate through the winter months?
    Useful resources

    Make your revision easier

    Maths

    Maths Forum posting guidelines

    Not sure where to post? Read the updated guidelines here

    Equations

    How to use LaTex

    Writing equations the easy way

    Student revising

    Study habits of A* students

    Top tips from students who have already aced their exams

    Study Planner

    Create your own Study Planner

    Never miss a deadline again

    Polling station sign

    Thinking about a maths degree?

    Chat with other maths applicants

    Can you help? Study help unanswered threads

    Groups associated with this forum:

    View associated groups
  • See more of what you like on The Student Room

    You can personalise what you see on TSR. Tell us a little about yourself to get started.

  • The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

    Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

    Quick reply
    Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.