I am trying to calculate a loan amount where the total principle depends on the payment amount.

Here's a simple example, leaving aside that a) no bank would actually do this, and b) my income is not specified:

I have `$0` in the bank right now. I want to take out a loan for a `$100` car, for which I will be making a regular annual payment, starting at the end of year 0 with an interest-only payment. My loan rate is 5%. In order to qualify for the loan, the bank wants to make sure that I have a balance equal to twice my annual payment amount throughout the life of the loan.

If I borrow exactly `$100`, my payment at the end of year 0 would be `$5` (`$100` x `5%`). However, I need to borrow a little extra so that I have a `$10` balance in the account (to satisfy the requirement that I maintain a balance of twice my annual payment). This means I actually need a loan of `$110`. However, a loan of `$110` actually raises my payment amount to `$5.50`, which means I need to maintain a balance of `$11` in my account.

This means that I actually need to borrow `$111` to begin with, which raises my payment to `$5.55`, which means I actually need to borrow `$111.10`, etc.

I have worked it out to a summation that resembles the following, although I can't be certain this is accurate:

$$\sum\_{n=1}^x (x-1-n)r^n$$

Where $r$ is the interest rate

I believe this would allow for me to iterate $x$ number of times over a formula that sums the additional principle needed for each incremental amount of interest. However, I'm not sure how to carry this out to infinity.

Ultimately, I need to plug this into Excel as part of a cash flow, but I wanted to think through the math first.

Any ideas?