Today we worked on a math problem from the year 1202:
Suppose a newly-born pair of rabbits, one male, one female, are put in a field. The rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. How many pairs will there be in one year?
The answer was 144. The trick was to develop a good system for keeping track of newborn rabbits, maturing rabbits, and breeding rabbits. Once we did that (see someones notes or better yet, try it yourself) for the first few months, we generated a sequence of numbers. The domain of the sequence was the month number and the range was the number of pairs of rabbits. Sofia explained the pattern in the sequence and we found the answer to the question (144) without keeping track of all the rabbits.
The difficult thing about the rabbit sequence is that it is not arithmetic or geometric. We could not use the simple rules that we had learned. Instead we needed a new tool:
The recursive definition of a sequence.
See the slides for more info on this topic.
The answer to the question on the first slide is 511. The second slide shows an arithmetic sequence. We could write a simple formula using the general arithmetic sequence formula (do you know it?) but we can also write a recursive formula. Sometimes we won't be so lucky and the recursive definition will be the only one we can find.
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New Homework: Read pg. 470
On pg. 481 do problems 1-19 odd
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