# Lucas meets Fibonacci

Calculus Level 3

The $$n$$th Lucas number is defined such that $$L_n = L_{n-1} + L_{n-2}$$, where $$L_1 = 1$$ and $$L_2 = 3.$$
Similarly, the $$n$$th Fibonacci number is defined such that $$F_n = F_{n-1} + F_{n-2}$$, where $$F_1 = 1$$ and $$F_2 = 1.$$
Let $$Q_n$$ be the ratio $$\frac{L_n}{F_n}.$$ With simple calculation we find that $$Q_1 = \frac{1}{1} = 1,\ Q_2 = \frac{3}{1} = 3, ...$$

Find $\lim_{n\to\infty} Q_n.$

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