# Sequences and Roots!

Calculus Level 5

$\displaystyle P(x) = \sum_{m=0}^n F_mx^m$

Let $$P(x)$$ be an $$n^{th}$$ degree polynomial defined as above for a certain odd natural number $$n$$. And denote by $$\{r_1,r_2,r_3,\dots, r_n \}$$ as the set of roots of $$P$$.

Let $${a_k}$$ be a sequence defined by:

$\displaystyle \frac{1}{a_k-r_ka_{k-1}} = r_{k}r_{k+1}\dots r_n$

$$\forall k\leq n , k\in N^*$$, with $$a_0=1$$

Find:

$\displaystyle \lim_{n\to \infty} (a_n+1)F_{n+1}$

$$\mathbf{\; Details \; and \; Assumptions \; :}$$

$$\bullet \{F_m\}$$ denotes the Fibonacci Sequence ($$F_0=F_1=1$$).

$$\bullet$$ The set of roots of $$P$$ also takes into consideration the multiplicity (for example, double roots exist twice).

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