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Math Help - Find x and y if the sequence is geometric

  1. #1
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    Find x and y if the sequence is geometric

    Hello...
    Find x and y if the sequence: 2y, 2xy, 2, xy/2 is geometric

    Thank you for the help
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  2. #2
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    Re: Find x and y if the sequence is geometric

    if series is geometric
    there is a common ratio r such that
    a,ar,ar^2,... represent geometric series
    now here r=2xy/2y=2/2xy
    also 2/2xy=(xy/2)/2
    solving both these u get
    x=1/2 and y=4
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  3. #3
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    Re: Find x and y if the sequence is geometric

    Quote Originally Posted by IStandAlone99 View Post
    Hello...
    Find x and y if the sequence: 2y, 2xy, 2, xy/2 is geometric

    Thank you for the help
    Let's take this in pieces. What is the general formula for a geometric series? How many variables does that general formula have?

    prasum beat me to it.
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  4. #4
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    Re: Find x and y if the sequence is geometric

    Hello, IStandAlone99!

    \text{Find }x\text{ and }y\text{ if the sequence: }\,2y,\,2xy,\,2,\,\tfrac{xy}{2}\text{ is geometric.}

    We have: . \begin{Bmatrix}a &=& 2y & [1] \\ ar &=& 2xy & [2] \\ ar^2 &=& 2 & [3] \\ ar^3 &=& \frac{xy}{2} & [4] \end{Bmatrix}

    \begin{array}{ccccccccc}[2]\div[1]: & \dfrac{ar}{a} &=&  \dfrac{2xy}{2y} & \Rightarrow & \;\;r &=& x & \; [5] \end{array}
    \begin{array}{ccccccccc} [3]\div[2]: & \dfrac{ar^2}{ar} &=& \dfrac{2}{2xy} & \Rightarrow & r &=& \dfrac{1}{xy} & [6]\end{array}
    \begin{array}{ccccccccc} [4]\div[3]: & \dfrac{ar^3}{ar^2} &=& \dfrac{\frac{xy}{2}}{2} & \Rightarrow & \;\;r &=& \dfrac{xy}{4} & [7] \end{array}

    \begin{array}{ccccccccc}\text{Equate [7] and [6]:} & \frac{xy}{4} \,=\,\frac{1}{xy} & \Rightarrow & x^2y^2 \:=\:4 & [8] \\ \text{Equate [5] and [6]:} & x \,=\,\frac{1}{xy} & \Rightarrow & x^2y \:=\:1 & [9] \end{array}

    [8] \div [9]: \;\;\frac{x^2y^2}{x^2y} \:=\:\frac{4}{1} \quad\Rightarrow\quad \boxed{y \,=\,4}

    Substitute into [9]: . x^2(4) \,=\,1 \quad\Rightarrow\quad x^2 \,=\,\tfrac{1}{4} \quad\Rightarrow\quad \boxed{x \:=\:\pm\tfrac{1}{2}}


    Edit: Too slow again!
    I believe my solution echoes parsum's ... just more details.
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