recursion – What is a loop mathematically?

Here is a word problem

For every dollar Ben spent on bagels, David spent 25 cents less. Ben
paid $12.50 more than David. How much did they spend in the bagel store
together?

My programmer mentality said, Oh, this is a loop. Just loop on

x - .75x

incrementing x each step until the accumulator is ~ 12.50. That turns out to be x = 50. Then work backwards to get the other part, etc…

I know there is a close connection between recursion and math’s induction, but what have I got here mathematically? Is this a limit problem? It also looks like a fixed point issue. I don’t know enough about Analysis and series, but is this a type of series? In general, what part of math is a loop going toward an “eplison”?