Sequence problem

  • Thread starter Couperin
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  • #1
Couperin
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Homework Statement

A country has a stable population of 60'000'000 people.
The country can be divided into two regions - a prosperous North and a poorer South.

Currently, 20'000'000 people live in the North while 40'000'000 live in the South.

A model for population movement predicts that each year:
8% of the people living in the South move to the North, and
2% of the people living in the North move to the South.

Let [tex]p_n[/tex] denote the population of the Northern part of the country in [tex]n[/tex] years' time.

Prove that [tex]p_n[/tex] satisfies
[tex]p_0 = 20000000[/tex], [tex]p_n+1 = 4800000 + 0.9p_n[/tex]

My problem

There are some questions involved with this question, and I've answered them. What's bugging me is, I'm sat there looking at this sequence equation, wondering how on Earth the examiner managed to derive it. Could anyone tell me how?

I've tried all sorts of stuff to figure out where the number 4800000 came from, but to no avail.

If it's any use, here's some stuff I managed to find.

[tex]NPop_n+1 = 0.98NPop_n + 0.08SPop_n[/tex]
[tex]SPop_n+1 = 0.92SPop_n + 0.02NPop_n[/tex]
 

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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The population movement says N(n+1)=N(n)-0.02*N(n)+0.08*S(n). Since the population is stable S(n)=60000000-N(n). Substitute that into the first equation.
 
  • #3
Couperin
57
0
Thanks for your help.
 

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