Question
It's the $4$th of July, so to be very patriotic, you decide to invest $\$b_0$ in a stock today that doubles in value each day, and then drops the total account value by $\$r$ immediately after each doubling. For example, if $r = 40$ and $b_0 = 100$, you would have $\$160$ in your account tomorrow. You find that after $n$ days, you have $\$0$ in your account. How much money was in the account $0 \leq k \leq n$ days after your initial investment? Report the answer with $r = 1280, n = 8$, and $k = 5$.