statistics – Average number of attempts to success with increasing probability

I am struggling to solve the following problem:

Given the opportunity for base success $$P_0$$ = $$s$$ and increasing the probability of success in each failed test (in my case, I guess a simple constant increase $$d$$: $$P_n$$ = $$s$$ + $$n$$*$$d$$), find the expected average number of trials $$E (n)$$.

Probability of success in $$n$$-the essay could be expressed as $$(1 – P_1) * (1 – P_2) * .. * P_n$$ (Think of all previous attempts that are negative since we are looking for the first successful test). However, I'm not sure how to find $$E (n)$$ besides listing all the possibilities (highly unwanted!).

Also, how would the solution for the generalized model with function change? $$D (n) = P_n$$ Expressing probability of each trial? (Leave $$D (n)$$ be geometric progression or $$sin (n * Pi / 6)$$ LOL)