MIT6_042JS10_lec32_prob - Massachusetts Institute of...

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Unformatted text preview: Massachusetts Institute of Technology 6.042J/18.062J, Spring 10 : Mathematics for Computer Science April 26 Prof. Albert R. Meyer revised April 22, 2010, 667 minutes In-Class Problems Week 12, Mon. Problem 1. The famous mathematician, Fibonacci, has decided to start a rabbit farm to fill up his time while hes not making new sequences to torment future college students. Fibonacci starts his farm on month zero (being a mathematician), and at the start of month one he receives his first pair of rabbits. Each pair of rabbits takes a month to mature, and after that breeds to produce one new pair of rabbits each month. Fibonacci decides that in order never to run out of rabbits or money, every time a batch of new rabbits is born, hell sell a number of newborn pairs equal to the total number of pairs he had three months earlier. Fibonacci is convinced that this way hell never run out of stock. (a) Define the number, r n , of pairs of rabbits Fibonacci has in month n , using a recurrence rela- tion. That is, define r n in terms of various r i where i < n . (b) Let R ( x ) be the generating function for rabbit pairs, R ( x ) ::= r + r 1 x + r 2 x 2 + . Express R ( x ) as a quotient of polynomials. (c) Find a partial fraction decomposition of the generating function R ( x ) ....
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MIT6_042JS10_lec32_prob - Massachusetts Institute of...

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