Finding the initial payment that is exponentially increasing each month. For example: After $12$ months, I want to have paid a total of $\$100,000$. Each month I want to increase my payment by $12\text{%}$ over the previous month. (1.12x)
How would I go about finding the first month's payment so that at the end of $12$ months, all the payments add up to that $\$100,000?$
Using excel's goal seek I would find that the answer is: $\approx\$4143.68,$ but I'm trying to find a way to write a formula that would solve that immediately instead of having to rely on Goal Seek or Solver.
Then, how would one go about fixing that formula based on different percentages and total sums?

You use the formula for the partial sum of the geometric series. Basically you want to solve the following equation:
$$x\cdot \sum_{i=0}^{11} 1.12^i =100000,$$
where $x$ is the the first month's payment. The term $\sum\limits_{i=0}^{11} 1.12^i $ can be simplified to $\frac{1.12^{12}-1}{0.12}$. Therefore
$$x=100000\cdot \frac{0.12}{1.12^{12}-1}$$
