Question

Brennan has been playing a game where he can create towns and help his empire expand. Each town he has allows him to create 1.15 times as many villagers. The game gave Brennan five villagers to start with. Help Brennan expand his empire by solving for how many villagers he can create with 15 towns. Then explain to Brennan 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

This is simple: You have a geometric progression with $a = 5$ and $r = 1.15$. The sum of the first $n$ terms of this progression is $ s_n = 5\left( \frac{1-(1.15)^n}{1-1.15}\right)$.

Answer 2

The answer to the first part of the question is $ floor(s_15 = 5\left( \frac{1-(1.15)^{15}}{1-1.15}\right)) = floor(47.58) = 47$ villagers.

Answer 3

To find the answer to the second part, you know that there are $s_n$ villagers. To find the number of towns you have to assume hat the maximum number of villagers has been created and rearrange $ s_n = 5\left( \frac{1-(1.15)^n}{1-1.15}\right)$ to make $n$ the subject, a follows:

Answer 4

$$ 5(1-(1.15)^n) = (1-1.15)S_n = -0.15S_n \\ 5-5(1-(1.15)^n) = -0.15S_n \\ 5(1.15)^n = 5+0.15S_n \\ (1.15)^n = \frac{5+0.15S_n}{5} \\ S_n = log_{1.15}\frac{5+0.15x}{5} $$