Derive paying users over time with churn

Leaky-bucket model. Each month t: N new users start a free trial; a% convert to paid at month-end; b% of existing (start-of-month) paying users churn at month-end. Let P_t be paying users at end of month t with P_0 given. (i) Derive P_1, P_10, and a closed-form P_M in terms of P_0, N, a, b. (ii) Find the steady state P* and the condition for convergence/stability. (iii) Re-derive if churn also applies to same-month converters (i.e., churn applied to all payers at month-end). (iv) If N, a, b vary by month, write the general recurrence and state when P_M has a closed form.