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 $$d$$.

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