I play the following game using a coin that lands heads with probability $p$. I start with $X_0$ = $1$ and at each stage I gamble all I have on the toss of the coin. If it lands heads I end up with twice what I started with; if it lands tails I lose everything. All coin tosses are independent.
(1) With $X_n$ denoting how much money I have after the nth toss, find $E[X_{n+1}|X_n = k]$ in terms of k. (2) Find $E[X_n]$ for all $n$.
My attempt at solution:
(1) If we know that $X_n = k$, and the $n+1$ toss will be either H or T with probability $p$ and $1-p$ respectively, that means Xn+1 will be either 2k or 0 respectively. So $$E[X_{n+1}|X_n = k] = (p)(2k) + (1-p)(0) = 2pk$$
(2) Not sure if I have the right thought process but $E[X_n] = 2p(X_{n-1})$ because there's a $p$ probability of doubling $X_{n-1}$ and $1-p$ probability of getting $0$. So using the same logic as in (1), it would be $2p(X_{n-1})$.
I think my reasoning for (2) is correct, but I'm not sure if that's the final answer.