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

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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