This is used for calculating monthly costs that increase by a certain percentage month-over-month when given a forecasted annual budget.
An annual increase of 15% annually or 1.28% monthly (equalling 15% annually) is expected.
Given that I know the target value (a sum of all the months) to be, $100,000, how would I calculate the initial value?
Final value: $100,000
Period growth rate: 1.28%
Periods: 11
Initial value: X
Is there a formula for this use case?
Not seeing a general formula in Gradshteyn and Ryzhik, etc. But it's easy enough to formulate generally: let $V_f=\$100,000$ be your final value, $p=0.0128$ your $1.28$% growth rate, $n=11$ your number of periods, and $V_0$ your initial value to be determined. Then $$V_f = \sum_{i=1}^n V_0(1+p)^{i-1} = V_0\sum_{i=1}^n (1+p)^{i-1}$$ so you just need a general formula for $\sum\limits_{i=0}^{n-1} x^i$ (where $x$ is our $1+p$), which is what I'm not seeing. But it's trivial to program, and given your postings on stackoverflow, I'd guess extremely trivial for you.