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    Could someone outline the proofs for:

    Quadratic equation
    Arithmetic Progressions (Series)
    Geometric Progressions (Series)

    Or alternatively link to sites which explain them. Thanks.
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    Arithmetic

    (a) + (a + d) + (a + 2d) + ... + (a + (n-2)d) + (a + (n-1)d)
    writen backward this is
    (a + (n-1)d) + (a + (n-2)d) + ...+ (a + 2d)+ (a + d) + (a)

    add these 2 together term by teram and you get
    (2a + (n-1)d) + (2a + (n-1)d) + .... + (2a + (n-1)d)
    n(2a + (n-1)d)
    this is twice the total so the total is
    (n/2)(2a + (n-1)d)
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    Geometric

    S = a + ar + ar^2 + ... + ar^(n-1)

    muliply this by r and you get

    rS = ar + ar^2 + ar^2 + ... + ar^(n)

    subtract the first from the second and you get
    S - rS = a - ar^n = a(1 - r^n) (as all the middle terms are present in both series they cancel)

    S(1-r) = a(1 - r^n)

    so S = a(1 - r^n)/(1-r)
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    Quadratic expression
    the quadratic expression proog is just completing the square on ax^2 + bx + c = 0

    x^2 + b/ax + c/a = 0

    (x + b/2a)^2 = -c/a + (b/2a)^2

    (2ax + b)^2= b^2 - 4ac (just muliplying through by 4a^2)

    2ax + b = +/-rt(b^2 - 4ac)

    2ax = - b +/-rt(b^2 - 4ac)

    x = [- b +/-rt(b^2 - 4ac)]/2a
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    Thanks
 
 
 
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Updated: April 14, 2006
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