Lawrence's sum

Algebra Level 3

If the sum

\[ \sum_{n=1}^{2111} \frac{n+2}{n! + (n+1)! + (n+2)!} \]

is written as \( \frac{1}{2} - \frac{1}{a!}\), what are the last three digits of \(a\)?

This problem is posed by Lawrence L.

Details and assumptions

The last three digits of \(1023\) is \(023\). You may enter your answer as \(023\) or \(23\).

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