Suppose the trader starts with an account balance of $\$100$. For each trade $n=1,2,3,...$ (each trade is closed with its profit/loss reflected immediately in his account balance before starting the next trade), he has a $1-w$ chance of losing $\$R_n$ and a $w$ chance of winning $\$kR_n$ where $k>0$ upon closing the trade, where $$R_n = \begin{cases} p\cdot \text{current account balance}, & n=1 \\ \text{max}(p\cdot\text{current account balance}, R_{n-1}), & n>1 \end{cases}$$ where $0<p<1$.
After each trade, if his account balance $\leq\$0$, he is unable to continue trading. What is the probability of his account balance reaching $\leq\$0$ before carrying out $N$ trades?
I am considering a specific case where $w=0.5, k=2, p=0.2, N=100$, which means the trader is risking $20\%$ of his account to have a $50\%$ chance of winning $40\%$ of his account. If he wins the trade, for the next trade he will risk $20\%$ of his new (greater) account balance, while if he loses the trade, he risks the same dollar amount as the previous trade. I have trouble understanding how to find out the probability of he blowing up his account before completing $100$ trades.