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Assume that rabbits do no reproduce during the first month of their lives but that beginning with the second month each pair of rabbits has one pair of offspring per month. Assuming that none of the rabbits die and beginning with one pair of newborn rabbits, how many pairs of rabbits are alive after \(n\) months?

There are three types of grocery stores in a given community. Within this community (with a fixed population) there always exists a shift of customers from one grocery store to another. On January \(1,1 / 4\) shopped at store \(\mathrm{I}, 1 / 3\) at store II and \(5 / 12\) at store III. Each month store I retains \(90 \%\) of its customers and loses \(10 \%\) of them to store II. Store II retains \(5 \%\) of its customers and loses \(85 \%\) of them the store I and \(10 \%\) of them to store III. Store III retains \(40 \%\) of its customers and loses \(50 \%\) of them to store I and \(10 \%\) to store II. a) Find the transition matrix. b) What proportion of customers will each store retain by Feb. 1 and Mar. 1 ? c) Assuming the same pattern continues, what will be the long-run distribution of customers among the three stores?