Bob J.

asked • 04/25/16

Recursive rule

A bird sanctuary initially contains 1000 birds. Each year, 10% of the birds fly away, so the sanctuary is restocked with 40 new birds. Write a recursive rule for the number an a n of birds at the start of the n n th year.

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Alan G. answered • 04/25/16

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You are given a1 = 1000. After (n-1) years, the number of birds decreases by 10% and 40 new birds are added, so the number at the start of the nth year is:
 
an = .9an-1 + 40.
 
Explanation: Since 10% are lost each year, 90% = .9 are left. .9an-1 is the number of birds from the last year times 90%.
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Bill P.

Another interesting question might be,"What is the minimum number of birds that will be present in the bird sanctuary at the end of a year after adding in the 40 new birds ?" Remember that you must round to the nearest whole number after 10% fly away.
 
I believe eventually there will be exactly 404 or 405 birds at the end of any year. From 404, 40 birds will fly away, but 40 new birds are added. I also believe that it is interesting to note that if you start with any quantity greater than 404, you will eventually have 404. In other words, the limit of this function when approaching it from the right is 404. However, when approaching from the left (begin with any number less than 390), the limit is only 395 or 396 depending on how you round. My point is under these conditions you can never have between 397-403 birds at the end of any year.
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04/29/16

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