# 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

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

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

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

Find the common difference \(d\).