# Cubing unities?

Geometry Level 3

${ S }_{ n }=\sum _{ i=1 }^{ n }{ \cot ^{ -1 }{ \left( { i }^{ 2 }+i+1 \right) } }$

Conside the summation above, $$S_n$$. If the value of $${ S }_{ 100 }$$ can be written as $$\cot ^{ -1 }{ \left( \frac { a }{ b } \right) }$$ for coprime positive integers $$a$$ and $$b$$ with $$a>b$$, find the value of $$2(a+b)$$.

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