stevens@teklabs.UUCP (09/30/83)
I like this kind of problem, let's get off this probability thing. Problem: Given Y(n) = Y(n-1) - X(n-1) X(n) = X(n-1) - X(n-1)*X(n-1)/Y(n-1) Find X(1) that maximizes X(8) when Y(1) = 4 OK, lets let X(1) = a for now. Through some calculations, Y(1) = 4 X(1) = a Y(2) = 4-a X(2) = a(4-a)/4 Y(3) = (4-a)^2/4 X(3) = a(4-a)^2/16 Y(4) = (4-a)^3/16 X(4) = a(4-a)^3/64 Y(n) = (4-a)^(n-1)/2^n X(n) = a(4-a)^(n-1)/4^(n-1) So to find the maximum of X(n) for any n, we take the derivative (sp?) with respect to a. We get dx/da = 0 = [a(n-1)(4-a)^(n-2) + (4-a)^(n-1)] or simply a(n-2) + 4 = 0 When n=8, a = -2/3 There's probably a more theoretical way of solving this, but this works, so I use it. By the way, a = -2/3 is a MINIMUM and not a maximum. I wish I knew how to find the maximum that was asked for in the problem -- Steve Silberberg