Weird Recursion-2

Calculus Level 4

A recurrence relation is given by \(b_{ 0 }=\dfrac { 7 }{ 2 } \) and \(b_{ n }=8-\dfrac { 7 }{ b_{ n-1 } } \).

If \(b_{ n }=\dfrac { A+B.{ C }^{ n } }{ A+B.{ C }^{ n-1 } } \) and \(\displaystyle \lim _{ n\to \infty }{ b_{ n } } = F \), where \(A,B\) and \(C\) are positive integers and \(\gcd(A,B)=1\) then find \(A + B + C + F\).

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