You are playing a gambling game with a probability of $p$ of winning between (0.5, 1). You start with $Y_0$ money. Each round, you bet a fraction $k$ of that money. If you win, you get all your money back plus your bet. If you lose, you lose the money that you bet.
What is $E[Y_n]$, or the expected amount of money after $n$ rounds?
I am able to get a recurrence relation for $E[Y_n]$ in terms of $E[Y_{n-1}]$, but I don't know how to completely simplify this in terms of $p$ and $k$.
So if you start with $Y_0$ and you bet $kY_0.$ If you win your get $kY_0$ so your total is $Y_1 = (1+k)Y_0.$ If you lose your total is $(1-k)Y_0.$ So the expected value is $$ E(Y_1) = (p(1+k) + (1-p)(1-k))Y_0.$$ Then for the next round you have $$ E(Y_2|Y_1) = (p(1+k) + (1-p)(1-k))Y_1, $$ so using the tower rule $$ E(Y_2) = E(E(Y_2)|Y_1)) = (p(1+k) + (1-p)(1-k))^2Y_0.$$ Continuing on, $$ E(Y_n) = (p(1+k) + (1-p)(1-k))^nY_0.$$