RMO 2014

Algebra Level 4

Let \(\left\{ { a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 },.........,{ a }_{ 2016 },.... \right\} \) be an arithmetic progression such that it has a common difference \(d\) and

  1. \[\large{\sum _{ i=1 }^{ 1008 }{ { a }_{ 2i-1}^{2} } =0}\]
  2. \[\large{\sum _{ i=1 }^{ 1008 }{ { a }_{ 2i }^{ 2 } }=2016 }\]

and for all \(k\), \(k\in { Z }^{ + }\)

\[\large{{ a }_{ k }+{ a }_{ k+1 }=1}\]

Find the common difference \(d\).


This question is a numerical version of a generalized problem that appeared in RMO 2014
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