Consider a modified gamblers ruin Markov Chain on the set of natural numbers {0,1,2,3...}. We start in state 1, and at each turn, for state i, we have a probability p to go to state i+1, and a probability q to go to state 0. Meaning that in this modification losing once means going bankrupt immediately, regardless of current state. I am interested in getting an expression for the Expected number of turns to hit state 0, starting at state 1. I tried using splitting and then ended up with a recurrence relationship I though I can solve k(i)=p*k(i+1) + 1, with k(0)=0, but the answers I get don't seem to make any sense (after trying special case p=q=1/2, the expectations are negative).
Could anyone please suggest an approach for solving this problem? Thank you very much, and any help is much appreciated.