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.