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Algebra Level 5

\[\large f\left( x \right) =7{ x }^{ 2017 }+{ x }^{ 2 }+x+5\]

Let \(f\left( x \right) \) be a polynomial having roots \({ x }_{ i }\) for \(i=1\) to \(2017\).Then find the value of

\[\large \frac { 1 }{ { 10 }^{ 6 } } \left( \sum _{ n=1 }^{ 2017 }{ \prod _{ i=1 }^{ 2017 }{ \left( { x }_{ i }+n \right) } } -\sum _{ k=1 }^{ 2017 }{ { k }^{ 2017 } } \right) \]

Details and Assumptions:

  • You can use calculator for lengthy calculations.

  • Write answer till 2 decimal places

Inspiration.

This problem is original,
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