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According to this source, there are 108 billion people who have ever been born. So, way more dead (101 billion) than alive (7 billion). In how many years will that no longer be true?

For the sake of simplicity (and this is drastically simplifying it), let's assume the world started with 1 person in "year 0" and keep the birth rate fixed throughout all time at 2%. So, we can represent the current population $P$ in year $n$ as $r^n$ (where r=1.02, for 2% growth), and the total population of all prior years as $\sum_{i=0}^{n-1} r^i$. We're looking for $n>0$ such that:

$$r^n \geq \sum_{i=0}^{n-1} r^i$$ $$\implies r^n \geq \frac{r^n - 1}{r-1} \text{(using the geometric series formula)}$$ $$\implies r^{n+1} - 2r^n + 1 \geq 0$$

How do we solve this equation for $n$?

jamaicanworm
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Exponential growth is scale-independent. It doesn't matter where you are on the curve.

If $n$ people are alive at any given moment and the population exactly doubles every generation, then there are the same number dead as alive; this is the convergent series

$n/2 + n/4 + n/8 +...$

If growth is slower then there are more dead, and if growth is faster then there are more alive.

In the real world it is more complicated because you have to take account of changes in both growth rates and the proportion of people who die childless. Various guesstimates have been made over the years, and they differ wildly. Forty years ago people thought there were more alive than dead, now we think it far the other way. In truth, nobody has enough historical data to make accurate statements about the past.

Technology has brought us one further principle: the past is no longer any guide to the future.

Guy Inchbald
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