Number Theory Help..?

First, you should mention that F is the fibbonacci(sp?) numbers. To get the limit, it is easy to see that F(n) / F(n-1) has a limit. So, let L be this limit. Then, observe that F(n + 1) / F(n) will have the same limit, so we can multiply F(n + 1) / F(n) and F(n) / F(n - 1) and take the limit to get L ^ 2. So, L ^ 2 is the limit of (F(n) + F(n - 1)) / F(n-1) = 1 + (F(n) / F(n - 1)), this has limit 1 + L. Thus, we obtain L ^2 - L - 1 = 0, using the quadratic formula L is (1 +/- sqrt(5)) / 2, since 1 - sqrt(5) is negative and each term is positive, we conclude that L = (1 + sqrt(5)) / 2; this version is much simpler than the one you were looking for.