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Algebra Level 4

\[\large\dfrac{3}{1!+2!+3!} + \dfrac{4}{2!+3!+4!} +\cdots+ \dfrac{2016}{2014!+2015!+2016!}\]

If the above expression can be represented as\[\dfrac{1}{k!} - \dfrac{1}{(k+a)!} \; ,\] then find \(\dfrac{a}{k}\).

Notation:

\(!\) denotes the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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