There is an online casino that has a promotion where you get a 50%
bonus on all of your deposits.  So if you deposit $50, you get a $25
bonus.  The bonus will be awarded as many times as you want, and you
can make a withdrawal at any time, but as soon as you make a
withdrawal you will not be eligible for a bonus for one week.  The
casino also removes all bonuses from your account when you withdraw.

Now there is a game at this casino in which the player will win even
money 49% of the time, and lose 51%.  The maximum bet size for the
game is $100.  If you were to deposit $67 and get a $33.50 bonus and
bet $100 in this game, then 49% of the time you would win $100
($200.50 - $67 deposit - $33.50 bonus) and 51% of the time you would
lose $67.  This would make the expected profit from the scenario
$14.83 ($100 x .49 - $67 x .51).  This part of the math is easy, but
it gets much more complicated when we add in more hands.  I can do it
out by hand for a little longer, but with my limited knowledge of
calculus it starts to get confusing pretty quickly.  If we were to
make $100 bets until we won $200 or went broke my math would look
something like this:
Lose first hand: 51%
Win first two hands: 24.01% (.49 x .49)
Win, lose, lose: 12.74% (.49 x .51 x .51)
win, lose, win, win: 6.00%
win, lose, win, lose, lose: 3.18%
win, lose, win, lose, win, win: 1.50%
And so on.  

So our chance of winning like this is about 32%.  The expected profit
is now (32% x $200) - (68% x $67) = $18.44.  The expected profit goes
up when you take higher risks.  Since you cannot get a bonus for a
week after making a withdrawal, the goal is to make as much money as
possible each week.  We will always be making $67 deposits, and always
be betting $100 per hand on this game where we win 49% of the time. 
When we try and turn our $100 into $200 we profit, and we profit even
more when we try and turn it into $300.  However, there will be a
point where trying to make an extra $100 will actually cause us to
lose money.  What I need to know is when this will occur, and the
expected profit for every $100 increment up until this point.  I'd
also be interested in the methodology used to solve it.

Thanks