Question

Brian has been playing a game where he can create towns and help his empire expand. Each town he has allows him to create 1.17 times as many villagers. The game gave Brian eight villagers to start with. Help Brian expand his empire by solving for how many villagers he can create with 16 towns. Then explain to Brian how to create an equation to predict the number of villagers for any number of towns. Show your work and use complete sentences.

Answer 1

84 villagers

Explanation:

This is a geometric sequence, so you would use the explicit formula of an = a1r^(n − 1)

n=16 and the r=1.17

Plug that into the equation and it reads a(16)=8*1.17^15 =84.309 and since you cant have a .309 of a villager you round down for a total of 84 villagers.

Answer 2

For the answer to the question above,
f(x) = k*b^x

k is the value of your thing (in this case number of people) after 0 iterations. b would be the rate that the thing grows, in this case by 17%. x is the number of iterations (towns) that have been expanded to, hence how many times 1.17 is compounded. So our equation would look like this:

f(x) = 8(1.17)^x

simply plug in the number of towns he has. So in this case 16.

f(16) = 8(1.17)^16

do the math (simplify)

f(16) = 98.64...

you can't have 0.64 of a person so we round down because that guy hasn't been "created" yet.

after 16 villages 98 people