User blog comment:QuasarBooster/Fibonacci/Lucas Sequence Extentions?/@comment-2033667-20150730151541

Actually, there's a simple proof with much lighter use of Binet's formula. Again define $$F_n = \frac{\varphi^n - \psi^n}{\varphi - \psi}$$. We will show by induction that $$S_a(n) = F_n + aF_{n-1}$$. The base cases are $$S_0(n) = F_0 + aF_{-1} = a$$ ($$F_{-1} = 1$$ per Binet's formula) and $$S_1(n) = F_1 + aF_0 = 1$$. Suppose $$S_a(n) = F_n + aF_{n-1}$$ and $$S_a(n + 1) = F_{n+1} + aF_n$$, then $$S_a(n + 2) = S_a(n) + S_a(n + 1) = F_n + aF_{n-1} + F_{n+1} + aF_n = F_{n+2} + aF_{n+1}$$. This completes the induction proof, so $$S_a(n) = F_n + aF_{n-1}$$.

Your constant, then, is $$\lim_{n \rightarrow \infty} \frac{F_n + nF_{n-1}}{F_{n-1} + nF_{n-2}}$$. I'm certain that this is equal to $$\varphi$$.