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# Calculus/Total fertility rate related mathematics

Question
Greetings!

Below I have linked an article from which I am deriving my question:
http://www.un.org/esa/population/publications/longrange2/WorldPop2300final.pdf

It states that with the population growth (total fertility rate) of 1995-2000 period which was around 6 at that time, the world population will be 244 billion in 2150 and in 2300 a meaty 134 trillion.

So my question is that if we halve the total fertility rate to around three to four children per woman, will that halve the population estimates at their estimated dates, I.E. 122 billion people in 2150 and 67 trillion in 2300?

I am writing a science fiction story of my own and I would like some help in terms of how many human we'd have with a total fertility rate of around three to four in the years 2150 and 2300.

So could you help me out here?

It would far more than half it, for it is multiplicative.  Assuming the families have an average of 4 children and it take roughly 25 years for each generation, the would give roughly 160 billion in 2150 and 10.5 trillion in 2400.  Assuming families only had 3 children, that would be 35 billion in 2150 and 400 billion in 2400.  Assuming families only had between 2 and 3 children each, that would give 13 billion in 2150 and 50 billion in 2400.

A good way to look at this that's simple and easy to understand, it consider the numbers 2 and 3.
2^2 is 4 whereas 3^2 is 9, so 3 produces over twice what 2 does when squared, which is equivalent to 1 period going by, for we start at period 1 and we've gone to period 2.
After 2 periods have gone by, that puts us in period 3.  Now 2^3 = 8 whereas 3^3 = 27.
After 5 periods have gone by, 2^6 = 64 whereas 3^6 = 729.  As can be seen, increasing the base by just a little bit increases the outcome by more each period.

Calculus

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