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)