It’ll overestimate by a lot if you do it over longer time periods. e.g. it overestimates this average by about 50% (your estimate actually gives 375, not 325), but if you went from 1m to 700m it would overestimate by a factor of about 3.
A pretty-easy way to estimate total population under exponential growth is just current population 1/e lifetime. From your numbers, the population multiplies by e^2.5 in 300 years, so 120 years to multiply by e. That’s two lifetimes, so the total number of lives is 700m2. For a smidgen more work you can get the “real” answer by doing 700m 2 − 50m 2.
That approximation looks like this
It’ll overestimate by a lot if you do it over longer time periods. e.g. it overestimates this average by about 50% (your estimate actually gives 375, not 325), but if you went from 1m to 700m it would overestimate by a factor of about 3.
A pretty-easy way to estimate total population under exponential growth is just current population 1/e lifetime. From your numbers, the population multiplies by e^2.5 in 300 years, so 120 years to multiply by e. That’s two lifetimes, so the total number of lives is 700m2. For a smidgen more work you can get the “real” answer by doing 700m 2 − 50m 2.