Question

Geometric Series help. *Question attached below* Will give medal.

Answer 1

So I'm trying to figure out how to start tackling this problem. I know how to do part a, but I am clueless as to how to begin. Since no value of what the population is was given?

Answer 2

well we could cheat for b

Answer 3

like we find the difference of p_ n and p_n+1 to get a recursive formula

Answer 4

\[p_{n+1}-p_{n}=[500(1.2)^{n+1}+500]-[500(1.2)^{n}+500] \\ p_{n+1}-p_n=500(1.2)^n[1.2-1] \]

Answer 5

Hmm, true!

Answer 6

It actually hard for me to see either the equation in B and the equation I just came up with from the information given.

Answer 7

I mean the equation in C and also the equation we came up with just now

Answer 8

This is more troublesome than I thought it would be

Answer 9

technically we could cheat for all this information
but I would like to actually understand the problem
like we could use C to find p1 and p2

Answer 10

and say we didn't

Answer 11

I don't get where the 500's come from?
like we don't know the initial population right?

Answer 12

\[.2 \cdot p_0+p_0 -100=p_1 \\ (1.2)p_0-100=p_1\]
this is what I thought would be p1
since the initial population p0 is increased by 20%
and so the new population would have been .2p0+p0 but then we had 100 leave at the end

Answer 13

I get that bit. But it said to calculate the value and I'm not sure how we're going to do that because no initial population number was given

Answer 14

maybe we can use differential equation
I was reading about growth-decay initial value problems here:
http://www.math.utah.edu/~gustafso/2250exp-modeling.pdf
on page 4

Answer 15

yes I think we will definitely need a differential equation
I see how they get the 500 now

Answer 16

like I have this
p'=rate in-rate out right?

Answer 17

tell me if you agree with this...
\[p'=(\text{ rate in })-(\text{ rate out})\\ p'=(.2p)-(100)\]

Answer 18

that is a linear first order differential equation

Answer 19

Yes, I agree with that :)

Answer 20

so we can solve that for p

Answer 21

But the constant of integration i'm still having trouble finding that

Answer 22

because I'm pretty we haven't been given a point on p

Answer 23

I agree..we weren't given any value of P.

Answer 24

Let me see if @Kainui knows... Trying to know how to find constant of integration when it doesn't seem any initial condition was given.

Answer 25

While it's true you weren't given an initial value, it does look like you were given an initial percentage. I think that should be enough though.

Answer 26

I still don't see how the initial percentage can help to solve the question :$

Answer 27

Is this all the information you were given? For instance are these assuming a certain kind of growth? What are the previous 3 questions like and is anything said concerning the whole problem set?

Answer 28

All I was given is that the population increases by 2% and 100 people leave the population at the end of each year

Answer 29

Are you sure you're not supposed to assume something like \[\Large P(t)=P_0e^{kt}\] kind of thing?

Answer 30

I'm not sure. We've actually never done this topic in class as yet. Monday we'll be doing it, but we have an assignment to complete on it

Answer 31

What class is this?

Answer 32

Pure Mathematics