Question

Word problem on Geometric Sequences?
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

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, a follows:

$$ 5(1-(1.15)^n) = (1-1.15)S_n = -0.15S_n \\ 5-5(1-(1.15)^n) = -0.15S_n \\ 5(1.15)^n = 5+0.15S_n \\ (1.15)^n = \frac{5+0.15S_n}{5} \\ S_n = log_{1.15}\frac{5+0.15x}{5} $$

Answer 2

\(\huge\color{red}{WELCOME~TO~OPENSTUDY}\)
http://w3facility.org/question/geometric-series-word-problem-help/

Answer 3

oh acha so this is how you do it?

Answer 4

yep :) seems you speak hindi :)

Answer 5

haha yes i do :-)
thanks for taking time out to help me

Answer 6

hehe your welcome :)

Answer 7

haha yes i do :-)
thanks for taking time out to help me

Answer 8

your welcome :)

Answer 9

@pooja195 so the equation is a prediction of how many villagers will be in each town?

Answer 10

Yes

Answer 11

Okay. Thank you!