The assumption is that into each generation is born a certain number of children, and those children have a certain number of children, etc. Gender is not being taken into account, and the children's spouses are not being tallied here. And no, this is not a worm population growth chart. I am just trying to get an idea of how many generations it would take to get to about 250 billion people born into it, give or take a couple billion. I haven't taken interbreeding into account either, as after about a 5 generation gap some consider it safe to interbreed though some consider sooner than that safe. Doing that however would add yet even more complexity that I haven't even tried to think about adding.

Look, if you know the generational growth rate (i.e. for each person in generation N, there are 2.5 people in generation N+1, or whatever), then just start with your 250 billion and work backwards. (2.5 is the average of rand(6).)

$n = 250_000_000_000;

$a = 2.5;

while ($n>2) {
  print "$n\n";
  $n /= $a;
}

^D
250000000000
100000000000
40000000000
16000000000
6400000000
2560000000
1024000000
409600000
163840000
65536000
26214400
10485760
4194304
1677721.6
671088.64
268435.456
107374.1824
42949.67296
17179.869184
6871.9476736
2748.77906944
1099.511627776
439.8046511104
175.92186044416
70.368744177664
28.1474976710656
11.2589990684262
4.5035996273705
[download]

There. 28 generations.

Approximately.

Or by modifying the compound interest formula to kill off the previous generation:

use strict;
use warnings;

my $startPop = 4;
my $targetPop = 250e9;
my $increase = 2.5;
my $n = int ((log($targetPop) - log($startPop)) / log($increase) + 0.5
+);
my $pop = int ($startPop * ($increase) ** $n);

print "$pop population after $n generations\n";
[download]

Prints:

222044604925 population after 27 generations
[download]

Add 1 instead of 0.5 in the $n calculation if you want a minimum of $targetPop.

True laziness is hard work