Step-by-step explanation:
so, there is a population in that planet. right ?
that means they discovered a populated planet.
anyway,
18 years
p0 is the start population.
p0 × 2 after 3 years
p0 × 2 × 2 = p0 × 4 after 6 years
p0 × 4 / 6 after 6 years
(p0 × 4 / 6) × 2 after 9 years
(p0 × 4 / 6) × 4 after 12 years = p0 × 8/3 after 12 years
(p0 × 8/3) / 6 after 12 years = p0 × 8/18 = p0 × 4/9
p0 × 4/9 × 2 after 15 years
p0 × 4/9 × 4 after 18 years
p0 × 4/9 × 4 / 6 after 18 years = p0×16/54 = p0 × 8/27 = 512
p0 = 512×27/8 = 1728
p(n) = p0 × (2/3)^(n/6)
with n being years.
because
the population changes by a factor 2 every 3 years = 2^(n/3)
and by a factor 1/6 every 6 years = (1/6)^(n/6).
so,
p(n) = p0 × 2^(n/3) × (1/6)^(n/6)
2^(n/3) = 2^(2n/6) = 2²^(n/6) = 4^(n/6)
therefore,
p(n) = p0 × 4^(n/6) × (1/6)^(n/6) =
= p0 × (4/6)^(n/6) = p0 × (2/3)^(n/6)
