A growth of triplets

Algebra Level 4

\(S=\dfrac { 1 }{ 1\cdot 2\cdot 3 } +\dfrac { 1 }{ 2\cdot 3\cdot 4 } +\dfrac { 1 }{ 3\cdot 4\cdot 5 } +\dfrac { 1 }{ 4\cdot 5\cdot 6 } +\dfrac { 1 }{ 5\cdot 6\cdot 7 } +\cdots\)

If \(S\) can be expressed in the form \(\frac { 1 }{ A }\) with \(A\) being a positive, composite integer, find \(A\).