Solve Recursive Series for Winning Game: A vs B

[8daysAweek]

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:

$$a_0 = 100$$
$$b_0 = X$$
$$a_n = a_{n-1}-{0.25}b_{n-1}$$
$$b_n = b_{n-1}-{0.5}a_{n-1}$$

But I got stuck here unable to solve the equations.

Any help or ideas will be appreciated._{*This is not homework}

 

[lanedance]

it may help to write it in matrix form
$$\begin{pmatrix}<br /> a_{n+1} \\ b_{n+1}<br /> \end{pmatrix}<br /> =<br /> \begin{pmatrix}<br /> 1 & -\frac{1}{4} \\<br /> -\frac{1}{2} & 1\\<br /> \end{pmatrix}<br /> \begin{pmatrix}<br /> a_{n} \\ b_{n}<br /> \end{pmatrix}$$

 

[lanedance]

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