First time poster, and, as my post will intimate, not a mathematician, just someone searching for answers. My question has two parts:
1) In a genealogical chart for a single individual starting with yourself and working backwards, you'll find a simple exponential trait to your preceding/antecedent group of ancestors, e.g.
you have one set of parents (2 people)
you have two sets of grandparents (4 people)
you have four sets of great-grandparents (8 people)
and so on...I'm only counting genetically-linked lineage (no step/half) for simplicity and using "sets" of ancestors rather than individuals.
However, whether you believe in Adam and Eve or in Darwin and Haldane, at a certain point all of this must converge back to an original set of antecedents (your common, original male/female ancestors, and logically the common human ancestors for all--the question I'll leave to philosophers and Richard Dawkins is how you get to a single ancestor not a single set of ancestors). Again, for simplicity, I'm only counting homo homo sapiens and not trying to take this back to the first unicellular organisms.
The question I'm trying to answer is, as one moves back in time, away from yourself (x=1) on a genealogical chart, your ancestors increase exponentially, but at some point they must start decreasing again to get back to a single set of common original ancestors (y=1)--for argument's sake, let's assume the decrease is perfectly proportional to the rate of increase and the time-series is based on finite generations not years--though if someone wants to try and model out interbreeding have at it.
When would this conversion/inflection across the generations need to occur--put another way, what is the maximum number of grand-nth parent sets you'd need to have before we started to see a need for this decrease--one would imagine it's about half-way back? In highly simple form it would go 1:2:4:2:1, but on a much grander scale.
There's an excellent article here from the BBC that talks about this issue as well as one known as the "genealogical paradox" (i.e. that most genealogy models show one to have more potential ancestors than human beings to have ever lived), and it also provides an important parameter for the time series: human history back to a single set of common ancestors for all humans is only about 3000 years or 100 generations. It also points out the need to assume inbreeding, consanguity, and incest as part of any genealogy, but for reasons both moral and mathematical let us keep things pure and simple.
2) The second part of my question pertains to the first: formulaically, how would one model out the math for the specific question above using the parameters described (e.g. 100 generations)? And, more generally, how would one write the formula for an exponential growth time series that starts at 1 and that must then suddenly inflect, and start to decay in proportion to its original exponential growth to ensure the final result is 1 at the end of the sequence? Put another way, what is the general formula for expressing a pattern that both increases and decreases across a time series such as 1:2:4:2:1 and could this be expressed in a single formula?
For bonus points: what fields of math are we discussing in this question and what would the graph for the specific ancestor and general formula equation look like? I believe in graph theory this is something called a directed acrylic graph?