The towns of Franklin and Chester each decided in the same year to begin posting their populations on signs by the main entrances into the towns. Each year, they update the signs.

Question

Franklin had a population of 20,000 people in the first year that they posted the sign, and each year, the population increases by approximately 5%

Chester had a population of 25,000 people in the first year that they posted the sign, and each year, the population increases by 500 people.

In which year will Franklin's population first exceed Chester's population?

Answer 1

you want the geometric progression and the arithmetic progression to be equal after n years. That is,

20000*1.05^(n-1) = 25000+500(n-1)
n = 8.4

so, 8.4 years after they started posting populations, the signs will read the same.
Thus, after 9 years Franklin's population will be greater.