# 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 $\sum _{ 1\le i\le j\le 100 }^{ }{ { r }_{ i } } { r }_{ j }.$

×