\(f(n)=\frac{1}{(x+n)(x+n+1)}\)

Algebra Level 2

\(f(n)=\frac{1}{(x+n)(x+n+1)}\) for a positive integer \(n\). If the following always holds regardless of the value of \(x\), what is the sum of the constants \(a+b+c\): \[f(12)+f(13)+f(14)+\cdots+f(19)=\frac{c}{(x+a)(x+b)}?\]

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