Starting with £k the gambler playes the game repeatedly until they either go bankrupt or reach £N, where N greater or equal to k.

Let \(\displaystyle B_{k}\) be the event that the gambler eventually goes bankrupt.

For k=1,2,...N-1 show that:

\(\displaystyle 3P(B_{k})=P(B_{k+1})+2P(B_{k-1})\)

I know the answer to this should be pretty simple, but I can't get it to work. A thorough solution would be of much help! Thanks