VWO 1997 Problem 3

$\Delta oa_1b_1$ is isosceles with $\angle a_1ob_1 = 36^\circ$ . Construct $a_2,b_2,a_3,b_3,...$ as below, with $|oa_{i+1}| = |a_ib_i|$ and $\angle a_iob_i = 36^\circ$ , Call the summed area of the first $k$ triangles $A_k$ . Let $S$ be the area of the isocseles triangle, drawn in - - -, with top angle $108^\circ$ and $|oc|=|od|=|oa_1|$ , going through the points $b_2$ and $a_2$ as shown on the picture. (yes, $cd$ is parallel to $a_1b_1$ there)

Show $A_k < S$ for every positive integer $k$ .