Let [tex]t[/tex] be the number of towns Chase builds. Each town contributes a factor of 1.13 to the number of villagers, so if [tex]V(t)[/tex] is the number of villagers, and Chase starts with [tex]V(0)=4[/tex] villagers, then
[tex]V(t)=4\cdot1.13^t[/tex]
For example, if he builds [tex]t=1[/tex] town, then his empire can support [tex]V(1)=4\cdot1.13=4.52\approx4[/tex] villagers. (Round down to the nearest villager.) Adding another town scales this up by 1.13, giving [tex]V(2)=1.13V(1)\approx5[/tex]. And so on.
Then with [tex]t=17[/tex] towns, he would be able to support [tex]V(17)=4\cdot1.13^{17}\approx31[/tex] villagers.
To explain this to Chase, you can describe recursively how each additional town affects the number of villagers:
[tex]V(1)=1.13V(0)[/tex]
[tex]V(2)=1.13V(1)=1.13^2V(0)[/tex]
[tex]V(3)=1.13V(2)=1.13^3V(0)[/tex]
and so on, giving the general rule
[tex]V(t)=1.13V(t-1)=1.13^tV(0)=4\cdot1.13^t[/tex]
same as we found earlier.
Your error is in thinking that you need to apply the geometric sum formula. It's not useful here.
