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Theorem [On Solving Certain Recurrence Relations] Consider the second order homogenous linear recurrence relation. f(n+1) +a f(n) + b f(n-1) = 0 where a and b are constants. (*) Consider also the quadratic equation x^2 + ax + b = 0. [This equation is called the characteristic equation of the relation (*).] CASE I: The characteristic equation has two distinct real roots, R and S. Then: (a) The function R^n is a solution to (*). (b) The function S^n is a solution to (*) (c) EVERY solution to (*) is of the form: P (R^n) + Q (S^n) for some constants P and Q CASE II: The characteristic equation has one repeated real root R. Then: (a) The function R^n is a solution (*) (b) The function nR^n is a solution to (*) (c) EVERY solution to (*) is of the from P (R^n) + Q (nR^n) for some constants P and Q. CASE III: The characteristic equation has two distinct complex roots u+iv and u-iv, where u and v are real numbers. Then: letting theta = arctan (v/u) and and letting T = square_root (u^2+v^2), (a) T^n sin (n theta) is a solution to (*) (b) T^n cos (n theta) is a solution to (*) (c) EVERY solution to (*) is of the form P [T^n sin (n theta)] +Q [T^n cos (n theta)] for some constants P and Q. Furthermore, there are no other possibilities. That is, every quadratic equation x^2 + ax + b = 0 either has two distinct real roots, one repeated real root or two complex roots of the form u+iv and u-iv. COROLLARY: The solution to the predator-prey equations are…… (we'll do that next).