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Post by vsiap on Nov 15, 2014 3:38:56 GMT
a) Since L n is the average of L n-1 and L n-2, the recurrence is L n = (1/2)L n-1 + (1/2)L n-2.  b) The characteristic polynomial x2 − x/2 − 1/2 = (1/2) (2x + 1)(x − 1) has the roots 1 and −1/2. Hence there exist real constants c 1 and c 2 such that Ln = c1 + c2(−1/2)n. The initial conditions imply the linear relations c 1 +c 2 = 100000 and c 1 −c 2/2 = 200000. The solution is c 1 = 500000/3 and c 2 = −200000/3, and we conclude Ln=500000/3+ (200000 · (−1/2)n)/3 . The second term converges to zero. Thus, the steady state scenario is that 166, 667 lobsters will be caught every year.
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