Well, the easiest way is to write a program or create a spreadsheet to answer the question. But if you insist on doing it using just math...
If the number of customers you acquire in the zeroth month is $C_0$, and the growth factor is $a$, then in month $n$ you acquire
$$C_n = a^nC_0$$
new customers. Therefore the total number of customers acquired is
$$C_{Tot} = \left( 1+a+a^2+\cdots+a^{11}\right)C_0 = \frac{a^{12}-1}{a-1} C_0$$
You said that $C_{Tot}\approx 100,000$ and $C_0\approx 2,500$, so the numbers work out if you have monthly growth in signups of about 20% (i.e. $a=1.2$).
You're assuming that each customer makes 4 payments of $10, one every 1.5 months, and then leaves. That means that once you've received all the payments, you expect to have received
$$$40 \times C_{Tot} \approx $4,000,000$$
However, the customers that arrive in the final month only make one payment, so you have to subtract
$$$30 \times a^{11}C_0 \approx $550,000$$
The customers arriving in month 10 only get a chance to make two payments (one when they arrive, one after 1.5 months) so you have to subtract
$$$20 \times a^{10}C_0 \approx $300,000$$
The customers in months 8 and 9 only get time to make three payments, so you have to subtract
$$$10 \times (a^8 + a^9) C_0 \approx $230,000$$
Therefore your total expected revenue after 12 months is around $2,900,000.