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Answer this question or recommend an answer
Numerical Solutions
an+1 = 0.9anFind an equilibrium value if one exists. Classify the equilibrium value as stable or unstable.
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- By Sharon |
- Category: other - mathematics |
- Grade level: college and above
- Aug/27/2008 |
- Answers to this question: 1
Answer by: Ganesha
First Answer
an+1 = (0.9)an = (0.9)(0.9an-1)=(0.9)2an-1= ---------=(0.9)na1 =(0.9)n+1a0, for integral n >= -1
This can also be written like,
an =(0.9)n*a0, for all integral n >= 0
Therefore,
an = (0.9)na0
Take a0 = 10000 as initial population. See its nth generation poulation.
You can see that the population of the next generation becomes 90% of its previous generation or dwindling down and sure to become extinct after 86 th genearation.
Sorry for showing partial tables as my internet bill becomes unbearable otherwise .
This can also be written like,
an =(0.9)n*a0, for all integral n >= 0
Therefore,
an = (0.9)na0
Take a0 = 10000 as initial population. See its nth generation poulation.
| Geration an | an =(0.9)*an-1 |
| a0 | 10000 |
| a1 | 9000 |
| a2 | 8100 |
| a3 | 7290 |
| a4 | 6561 |
| a5 | 5904.9 |
| a6 | 5314.41 |
| a7 | 4782.969 |
You can see that the population of the next generation becomes 90% of its previous generation or dwindling down and sure to become extinct after 86 th genearation.
| a80 | 2.184745005 |
| a81 | 1.966270505 |
| a82 | 1.769643454 |
| a83 | 1.592679109 |
| a84 | 1.433411198 |
| a85 | 1.290070078 |
| a86 | 1.16106307 |
| a87 | 1.044956763 |
| a88 | 0.940461087 |
| a89 | 0.846414978 |
| a90 | 0.76177348 |
| a91 | 0.685596132 |
| a92 | 0.617036519 |
| a93 | 0.555332867 |
Sorry for showing partial tables as my internet bill becomes unbearable otherwise .
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- Aug/30/2008 |
- Answers by Ganesha: 1193 |
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