Mathematics

The key here is exponential growth: when a quantity increases by a fixed percentage each time period, it multiplies by the same factor repeatedly. Rather than adding 12% of the original 500 each week (which would be linear), you're adding 12% of whatever you have, so the population compounds. To model this, use the form $P(t) = P_0(1 + r)^t$, where $r$ is the growth rate as a decimal and $t$ is the number of weeks—this captures how each week's growth builds on the last.

The key here is exponential growth: when a quantity increases by a fixed percentage each time period, it multiplies by the same factor repeatedly. Rather than adding 12% of the original 500 each week (which would be linear), you're adding 12% of whatever you have, so the population compounds. To model this, use the form $P(t) = P_0(1 + r)^t$, where $r$ is the growth rate as a decimal and $t$ is the number of weeks—this captures how each week's growth builds on the last.