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Algebra Level 5

In Fibonacci sequence, terms are defined $$\forall k \in \mathbb{N}, k \geq {2}$$, $$F_{1}:=1$$, $$F_{2}:=1$$ and $$F_{k+1}:=F_{k}+F_{k-1}$$. The number $$\phi$$ is given as the limit of the ratio between $$\frac{F_{k+1}}{F_{k}}$$, as $$k$$ increases indefinitely. Find the value of $$f(100)$$, where:

$$f(n)=\frac{cosh[(n+1)\cdot a]-cosh[(n-1)\cdot a]}{sinh(n\cdot a)}$$, $$a= ln \ \phi$$ and $$n \in \mathbb{R}-\{0\}$$.

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