Problem #40
Kyle's Mysterious Sequences
For a given integer n, Consider the following functions G(n) and F(n)
G(n) = \sqrt{F(n) - \frac{(n-1)^2}{n^2}F(n-1) + \frac{1}{n^2}}
F(n) = 8 + \frac{(n-2)^2}{n^2}F(n-2)
F(1) = F(2) = 8
Now consider the sequences
Y(n) = \sum{x} | x < n, n \mod x=0
Z(n) = Z(n-1) + Y(n)
Z(0) = Z(1) = 0
Find the value of Z(\frac{1}{(G (1000000000)-2)})