# A Golden Series

Algebra Level 4

$\large \sum_{n=1}^{\infty}\dfrac{F_{n}}{(\phi+1)^{n}}$

Let $$F_n$$ denote the Fibonacci number, where $$F_0 = 0, F_1 =1$$ and $$F_n = F_{n-1} + F_{n-2}$$ for $$n=2,3,4,\ldots$$.

If the value of the summation above equals $$s$$, find $$\lfloor 1000s \rfloor$$.

Details and Assumptions:

• $$\phi$$ denotes the Golden ratio, $$\phi = \dfrac{\sqrt5+1}2$$.

• $$\lfloor \cdot \rfloor$$ denotes the floor function.

• You may use the fact that $$\sqrt5 \approx 2.236$$.

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