Question

PLEASE HELP ME ALGEBRA 2

Answer 1

Let $t$ be the number of towns Chase builds. Each town contributes a factor of 1.13 to the number of villagers, so if $V(t)$ is the number of villagers, and Chase starts with $V(0)=4$ villagers, then

$V(t)=4\cdot1.13^t$

For example, if he builds $t=1$ town, then his empire can support $V(1)=4\cdot1.13=4.52\approx4$ villagers. (Round down to the nearest villager.) Adding another town scales this up by 1.13, giving $V(2)=1.13V(1)\approx5$. And so on.

Then with $t=17$ towns, he would be able to support $V(17)=4\cdot1.13^{17}\approx31$ villagers.

To explain this to Chase, you can describe recursively how each additional town affects the number of villagers:

$V(1)=1.13V(0)$

$V(2)=1.13V(1)=1.13^2V(0)$

$V(3)=1.13V(2)=1.13^3V(0)$

and so on, giving the general rule

$V(t)=1.13V(t-1)=1.13^tV(0)=4\cdot1.13^t$

same as we found earlier.

Your error is in thinking that you need to apply the geometric sum formula. It's not useful here.