Problem 475

Chase has been playing a game where he can create towns and help his empire expand. Each town he has allows him to create 1.13 times as many villagers. The game gave Chase 4 villagers to start with. Help Chase expand his empire by solving for how many villagers he can create with 17 towns. Then explain to Chase how to create an equation to predict the number of villagers for any number of towns. Show your work and use complete sentences.


number of villagers to start are 4
1.13 times increment in the next town,
So this is a geometric series example,
where a(1) = 4 and r = 1.13
we have a formula for sum of n terms in geometric series is :
S(n) = a(1)\frac{1 - r^{n}}{1-r}
so total number of villagers with 17 towns are :
S(17) = 4\frac{(1-(1.13)^{17})}{1-1.13}
S(17) = 4(\frac{-6989}{-0.13})
S(17) = 4(53.74)
S(17) = 214.96 ~ 215
So there are total 215 villagers with 17 towns.


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