So if they sell it back they do so before next fall?

So, let a_{n}, b_{n}, c_{n}and d_{n}be the number of new, second-hand, third-hand and fourth-hand books sold during fall in year n.

a_{0}= 5 million; b_{0}= c_{0}= d_{0}= 0.

a_{1}= 5 million - b_{1}; b_{1}= 0.9 a_{0}; c_{1}= d_{1}= 0.

a_{2}= 5 million - b_{2}- c_{2}; b_{2}= 0.9 a_{1}; c_{2}= 0.8 b_{1}; d_{2}= 0.

a_{3}= 5 million - b_{3}- c_{3}- d_{3}; b_{3}= 0.9 a_{2}; c_{3}= 0.8 b_{2}; d_{3}= 0.6 c_{2}.

Or eliminate b, c and d to get a_{n}in terms of a_{n-1}, a_{n-2}and a_{n-3}.

Make a spreadsheet from either to see it stabilise at a_{60}.

Homogenise the second as directed here Recurrence relation - Wikipedia, the free encyclopedia, and you can make a transition matrix,