Decision making quest. related with Markov chain or random walk(I guess?)

Question

I'm spending much of my time solving the following question.

I do understand some general strategy, which is comparing expected return for each turn, then deciding whether to go or not.

But this problem is too complicated for me. Can anybody help?

Thanks in advance!

*Oh and maybe, the following problem has a typo. $5 for every proceeding, not $3.

Here's a problem! Please click me.