Suppose we play a game where we start with $c$ dollars at each play you either double or halve your money, what is your expected fortune after $n$ trials?
My line of thinking went along the way that this is a Binomial distribution so my expected fortune will be $n/2$ at the end of the game. I couldn't verify this solution but I have a feeling that it might be wrong
ref: All of statistics by L.Wasserman
Update : Assume we start with c amount of dollars then after one game the expected amount of dollars we have is 5/4c , if we sum over a series of game we end up with n*(5/4)*c.
Fortune after n-th trials : $X_n=2^{\xi_n}2^{\xi_{n-1}}\ldots2^{\xi_1}c$ where $\xi_1,\xi_2,\ldots\xi_n\stackrel{iid}{\sim}Bernoulli(p)$ for $0<p<1$. So, $E(X_n)=E(2^{\xi_n}2^{\xi_{n-1}}\ldots2^{\xi_1}c)=E(2^{\xi_n})E(2^{\xi_{n-1}})\ldots E(2^{\xi_1})c=(E(2^{\xi_1}))^nc=(2p+(1-p)/2)^nc.$ If $p=1/2$ then $E(X_n)=(5/4)^nc.$