#2015_3 Not that easy, huh?

Algebra Level 4

If x1=2,x2=2+2,{ x }_{ 1 }=\sqrt { 2 } \quad ,{ x }_{ 2 }=\sqrt { 2+\sqrt { 2 } }, and likewise xn=2+2+2+.......n terms { x }_{ n }=\sqrt { 2+\sqrt { 2+\sqrt { 2+.......n\ terms } } } , then find 2sin(1+12+14+18+........+122006)45°2\sin\left( 1+\frac { 1 }{ 2 } +\frac { 1 }{ 4 } +\frac { 1 }{ 8 } +........+\frac { 1 }{ { 2 }^{ 2006 } } \right) 45°.

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