Inspired by Jake Lai

Algebra Level 3

Let \({ r }_{ 1 },\quad { r }_{ 2 },\quad { r }_{ 3 },...,\quad { r }_{ 100 }\) be the roots of the polynomial

\[\sum _{ k=0 }^{ 100 }{ { { \left( -1 \right) }^{ k }F }_{ k+1 }{ x }^{ 100-k } } ={ x }^{ 100 }-{ x }^{ 99 }+2{ x }^{ 98 }-...+{ F }_{ 101 }\]

Where \({ F }_{ n }\) is the \({n}^{th}\) number of the Fibonacci sequence.

Find the value of \[\sum _{ 1\le i\le j\le 100 }^{ }{ { r }_{ i } } { r }_{ j }.\]

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