# Fibonacci limit

Calculus Level 1

The Fibonacci sequence can be defined with induction as $$F_n=F_{n-1}+F_{n-2}$$, where $$F_1=0$$ and $$F_2=1$$. It is also well-known that the limit of the ratio between two consecutive terms $$\lim \frac{F_{n+1}}{F_n}$$ is the golden ratio $$\phi$$.

In the generalized Fibonacci sequence below, the original Fibonacci sequence is $$F_n^2.$$ $F_n^m=\sum_{i=1}^{m}{F_{n-i}^m}\ \text{ with }\ F_n^m=0\ \text{ if } \ n<m\ \text{ and }\ F_m^m=1.$ Every sequence will also have a generalized golden ratio $$\phi_m=\displaystyle \lim_{n \rightarrow \infty}{ \frac{F_{n+1}^m}{F_{n}^m}}.$$

What is the limit of the generalized golden ratio below? $\lim_{m \rightarrow \infty}{\phi_m} = \lim_{m \rightarrow \infty}{\lim_{n \rightarrow \infty}{\frac{F_{n+1}^m}{F_{n}^m}}}$

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