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.