Kaleidoscope Octagon

Geometry Level 5

A regular octagon \(ABCDEFGH\), which shall be named \(O_1\), has squares \(ACEG\) and \(BDFH\) inscribed in it. These squares form a smaller octagon, \(O_2\).

\(O_2\) 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 \(a+b\).

Hint: 200 follower special (Part 1).

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