Calculate ultimate survival when more than 1 survival curve is needed to determine outcome

Question

I would like to know if it is possible to combine multiple survival curves via an equation (e.g., via matrix multiplication or whatever) rather than stepping through multiple equations. E.g., assume you are a cable company and you have learned that new homeowners are great leads. As such, you subscribe to a service that gives you a list of new homeowners every month and you send them a postcard trying to get them to sign up for your service. Once you send your postcard, people will respond over time at the following rates:

A0  5%  # if you mail 1000 people, 50 will respond
A1  7%  # 7% of 950 remaining people (67 people) will respond
A2  3%  # 3% of 883 people will respond
A3  1%  # etc.

After customers sign up, some are happy with your service while others are not. As such, some percentage of people will leave while others stay. The likelihood of someone leaving follows the following curve:

B0   1%  # 1 out of 100 people that sign up for your service will stop in the 1st mo
B1   5%  # 5% of the 99 remaining will stop in month 2
B2   4%  # 4% of the remaining ~ 94 will stop in month 3
B3   3%  # etc.

So the question I'm trying to answer is: how can I calculate how many customers will I have in the month following the mailing (or whatever month) without performing each calculation by itself. Meaning, assuming I mail 1,000 customers, I will have 112.86 customers as of period 1 (where period 0 is the initial period / mailing period) calculated as:

1. 50 people responded the same month that I mailed (A0 * 1000), but 0.5 of them left in that same month (A0 * 1000 * B0) and another 2.475 left in the next month ((A0 * 1000) - (A0 * 1000 * B0)) * B1. This leaves me with 50 - 0.5 - 2.475 = 47.025 customers from the 1st tranche of responders.

2. 66.5 people signed up in the month subsequent to the mailing (1000 - (A0 * 1000)) * A1. But 0.665 of those customers will leave in the month that they sign up (67 * B0), leaving me with 65.835 customers from the 2nd tranche of responders.

So in total, I have 47.025 + 65.835 = 112.86 customers. I can obviously do the calculations to see how many customers I would have at any point in time, but it is tedious and (seemingly) over complicated. Right now I have a spreadsheet that does all the calculations, but if I want to do any intermediate step I have to do some manual intervention that I would rather not have to do (e.g., assume I mailed 1,000 customers in June and am now in August [period 2] and would like to know of the ~ 884 who have not responded yet, how many of them will be customers in November [period 5]?).

Sorry for the long winded question, but hopefully someone can help! Thanks in advance.