# 100-day streak!

**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:

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