# Kaleidoscope Octagon

**Geometry**Level 5

\(O_2\) has squares inscribed in it in the same manner, and the resulting hexagon formed is named \(O_3\). This pattern is repeated until infinity.

Let the area of octagon \(O_i\) be denoted by \(A_i\). Given that \(A_1=1\), for some integers \(a\) and \(b\), where \(a\) is square-free, \[\large \sum _{ i=1 }^{ \infty }{ { A }_{ i } } =\sqrt { a } +b\] Find \(a+b\).