Question

Need help please!

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

Just need someone to walk me through what I should do.

Answer 2

still need help

Answer 3

Yeah

Answer 4

ok i got u

Answer 5

so are u trying to work this out

Answer 6

y is how many villagers he will get with each town.
x is the number of the town
example: town # 7 y=1.15x7+5 this would mean that y=8.05

Answer 7

nice job

Answer 8

but i could have done tht

Answer 9

@alivejeremy Me?

Answer 10

lol

Answer 11

yea u

Answer 12

@sparklyme

Answer 13

@sparklyme You just got that off of Yahoo Answers, and anyways I need to know how to do this -- not the answer.

Answer 14

lol

Answer 15

Yahoo

Answer 16

lol

Answer 17

@Michele_Laino Can you help me out?

Answer 18

sure

Answer 19

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)Sn=−0.15Sn5−5(1−(1.15)n)=−0.15Sn5(1.15)n=5+0.15Sn(1.15)n=5+0.15Sn5Sn=log1.155+0.15x5

Answer 20

u get it

Answer 21

i tryed to break it down

Answer 22

That's not helping, don't copy and paste answers from other people's responses.

Answer 23

no i didn't someone help me wit tht question before

Answer 24

K12

Answer 25

It's obviously someone else's answer, I just looked it up.

Answer 26

ik u did

Answer 27

but tht person help me to

Answer 28

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 29

hint:
if we start wit \(a_1=5\) villagers, then after the first town we have \(5 \cdot 1.15=5.75\) villagers, namely \(a_2=1.15a_1\) villagers

Answer 30

i copy and paste tht one tho

Answer 31

at the next step, we have \(a_3=1.15 a_2\) villagers

Answer 32

@Michele_Laino dude u always give hints

Answer 33

he want u to break it down

Answer 34

I'm still not really understanding, am I supposed to plug in the first term and ratio of 1.15 into a geometric formula?

Answer 35

lol

Answer 36

no i didn't someone help me wit tht question before

Answer 37

hint:
in other words, we get the subsequent sequence:
\(a_1,\;a_2=1.15a_1,\;a_3=1.15a_2,\;...\)
and so on...

Answer 38

of course \(a_1=5\)

Answer 39

So 15 towns is a_1 [or 5] multiplied by 1.15 15 times?

Answer 40

yes! That's right!

Answer 41

it is a geometric sequence

Answer 42

Which would be 86.25? So 86.25 villagers can be created with 15 towns

Answer 43

the first term is \(a_1\), whereas the last term, after 15 towns, is \(a_{16}\), so we can write this:
\[\Large {a_{16}} = {a_1} \cdot {1.15^{15}}\]

Answer 44

since I have applied the general formula:
\[\Large {a_n} = {a_1}{q^{n - 1}}\]
and \(q=1.15\)

Answer 45

So an equation to find the nth amount of villagers for any amount of towns would be a(n) = a*1.15^n-1?

Answer 46

yes! with \(n=16\)

Answer 47

Ohh okay. So the first part of the question can be answered: 86.25 villagers can be made from 15 towns because a16=a1*1.15^15

Answer 48

and the second part can be written as: a(n)=a*1.15^(16-1) is the expression that can be used to find the nth term in this geometric sequence... or something like that?

Answer 49

we can write this:
\[\huge {a_n} = {a_1}\cdot {1.15 ^{(n - 1)}}\]

Answer 50

with \(a_1=5\)

Answer 51

Okay, I think I understand it now, thanks

Answer 52

:)