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Recursive Serieses

  1. Sep 1, 2010 #1
    There is a game with two players: A and B.
    Each turn the players shoot at each other simultaneously.
    Player A has 100 life points and the damage he inflicts is 50% of his remaining life points. Player B deals 25% respectively. Life points are rational numbers.
    A player wins the game when his life points are higher than 1, while his opponent's life points are smaller than 1.
    Find the minimum, natural starting life points that player B should have in order to win the game.



    I decided to start by representing the life points of each player as a series. I got this:

    [tex]a_0 = 100[/tex]
    [tex]b_0 = X[/tex]
    [tex]a_n = a_{n-1}-{0.25}b_{n-1}[/tex]
    [tex]b_n = b_{n-1}-{0.5}a_{n-1}[/tex]

    But I got stuck here unable to solve the equations.

    Any help or ideas will be appreciated.





    *This is not homework
     
    Last edited: Sep 1, 2010
  2. jcsd
  3. Sep 1, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    it may help to write it in matrix form
    [tex]
    \begin{pmatrix}
    a_{n+1} \\ b_{n+1}
    \end{pmatrix}
    =
    \begin{pmatrix}
    1 & -\frac{1}{4} \\
    -\frac{1}{2} & 1\\
    \end{pmatrix}
    \begin{pmatrix}
    a_{n} \\ b_{n}
    \end{pmatrix}
    [/tex]
     
  4. Sep 1, 2010 #3

    lanedance

    User Avatar
    Homework Helper

    then maybe examine the form of the matrix for several rounds, starting at ao, bo
     
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