Given a growing sequence of values, where each next element in the sequence is the prior value times a constant growth factor, what formula allows the computation of the starting value in order to reach a given target total of all elements in the sequence?
For example, if a sequence of integers grows by 20%, and you have six items in the sequence and your goal is a sum of 1,000, then the answer, rounding each value to the nearest integer, is between 100 and 101:
100, 120, 144, 173, 207, 249 = 993.
101, 121, 145, 175, 209, 251 = 1002.
I imagine there is a name for this kind of calculation and that it would relate to the kinds of formulas used in retirement savings, only it doesn't seem to quite match up since the growth is compounding on the stream, and not compounded also on the running sum. A floating point computation is good enough for estimating these integer values.
This kind of calculation (adding up all the terms of the series, not just computing the last term) actually does occur quite a bit in financial calculations.
The general formula (cribbed from https://math.stackexchange.com/a/636313, but it's a well-known standard formula) is $$ S = a + ar + ar^2 + \ldots + ar^{n-1} = \frac{ a( 1- r^n) } { (1-r) }, $$ provided that you do not round your terms.
For 20% growth you want $r=1.2$, and you have six terms in your sequence so $n=6$. You know the desired sum $S$; the only unknown is $a$. Solve for $a$, and that's the first term of the desired series (disregarding the effects of rounding).
Rounding each term to the nearest integer makes the problem quite a bit more complicated. The best bet is probably to use the formula without rounding, giving you some value of $a$, not necessarily an integer; then do trial and error, guessing integers that are close to the value of $a$ from the formula and summing the (rounded) values of each term of the series for each guess.