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 5 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

As it compounds, we can clearly see that every time he creates a town, the number of people increase by 1.15 times. so he creates a total of 15 towns, and he has 5 villagers to start with. So you would have to multiply it like ((5*1.15)) for the first village, and hten (5*1.15)*1.15 for the second village and so on. Therefore for 15 villages, it has to be (5*(1.15^15)), which if you have a calculator, is 40.6853081458. :P
Therefore its quite obvious that to create more villages, the equation would have to be (5*(1.15^n)), where n is the number of villages being made one after the other. 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)$. 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. 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. basically the equation is a prediction of how many villagers will be in each town. Hope this helps, have a nice day. Tag me on anymore you need help with.

Answer 2

Omg you're an angel im crying i love u so much bless ur soul @datchinookguy

Answer 3

@mariemiranda0314 lol your welcome :P

Answer 4

Let me know if you need help on anymore, just do tag me and if im not there in under five minutes im probably busy :/