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Math Help - difficulty to simplyfy a sum expression

  1. #1
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    Smile difficulty to simplyfy a sum expression

    Hello,

    I am trying to find the sum of a sequence, the problem is already solved in the book but not in a detailed way. While I understand the sum calculation process I don't understand the simplification result. I have marked in red the difference in the solution. In particular it says:

    etc etc
    Sum =
    (na/2) - (a/2)*(g^1 + g^2 + ... + g^n) + (nb/2) + (b/2)*(g^1 + g^2+ ... + g^n) =

    =(n/2)(a+b) + [(1-g^n)/(1-g)]*[(b-a)/2]
    the end

    My result is this:
    etc etc
    =(n/2)(a+b) + [(g-g^n+1)/(1-g)]*[(b-a)/2]
    the end

    I thought it could be some error, but there is the same result in an older version of the book. It might be something easy but still I cannot configure it out.

    Thank you all in advance,
    Melsi
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  2. #2
    MHF Contributor
    Opalg's Avatar
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    If you have transcribed this correctly then you are right and the book is wrong.
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  3. #3
    Junior Member
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    solved: thank you

    Thank you very much for your kind a quick reply! It's of help to have a second opinion.
    I double checked again that I have write correctly the book's solution here...I worked on it a lot and I am coming to same conclusion as you did:

    The particular section, how solved it (very detailed):
    (g^1 + g^2 + ... + g^n) =
    [(...)*(1-g)] / [1-g] =
    [(...)(1) + (...)(-g)] / [...] =
    [(...) + (-1)(g)(...)] / [...] =
    [(...) + (-1)(g^2 + g^2 +...+g^n + g^n+1)] / [...] =
    [(...) + (-g^2 - g^3 - ... -g^n - g^n+1)] / [...]=
    [g^1 + g^2 + ... + g^n -g^2 -g^3-...-g^n -g^n+1] / [...]=
    [g^1 -g^n+1] / [1-g] or [g -g^n+1] / [1-g] .

    Thank you Plato,
    [If you haven't done already I would suggest you read the Socrates apology saved-written by Plato! Socrates reasoning, the way he acts in the court is exceptional.]
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