After filing Form N-400, Application for Naturalization, the entire naturalization process can take anywhere from 5-8 months. This is an approximation. It may be shorter for some and longer for others.
Best Answer Thanks! I assume from your question that you have a court order for child support that ordered your ex to pay the $820 per month. Assuming this to be the case, and assuming that the court did not allow your ex to reduce his payment, you can go back 10 years to collect arrearages.
Depending on your state -- Texas for example -- you might have to pay back at least four years' worth of support, even if you can prove you didn't know you had a child. If your son's mother can prove you knew for a certain number of years that you had a child, a court can order back child support for all those years.