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Math Help - An irrational sequence

  1. #1
    Senior Member I-Think's Avatar
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    An irrational sequence

    A sequence of real numbers (u_1,u_2,u_3,...,u_n) has the following properties

    u_1= \sqrt{2} , u_2=\pi

    u_n=u_{n-1}-u_{n-2} for n\geq{3}

    What is the value of u_{2008}?

    Thanks for the help.
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  2. #2
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    Hello, I-Think!

    A sequence of real numbers \{u_1,u_2,u_3, \hdots u_n\} has the following properties

    u_1\:=\:\sqrt{2}
    u_2\:=\:\pi
    u_n\:=\:u_{n-1}-u_{n-2}\:\text{ for } n\geq{3}

    What is the value of u_{2008}?
    Crank out the first few terms . . .

    . . \begin{array}{ccc} u_1 &=& {\color{red}\sqrt{2}} \\ u_2 &=& {\color{blue}\pi} \\ u_3 &=& \pi - \sqrt{2} \\ u_4 &=& \text{-}\sqrt{2} \\ u_5 &=& \text{-}\pi \\ u_6 &=& \text{-}\pi + \sqrt{2} \\ u_7 &=& {\color{red}\sqrt{2}} \\ u_8 &=& {\color{blue}\pi} \\ \vdots & & \vdots \end{array}


    The sequence "loops" through a six-step cycle.


    \text{Since }2008 \:=\:334(6) + 4,

    . . \text{then: }\:u_{2008} \;=\;u_4 \;=\;-\sqrt{2}

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