By: Tao Steven Zheng (郑涛)

【Problem】

In "Quadrature of a Parabola", Archimedes (c. 287 - 212 BC) calculated the area segment of a parabola using the method of exhaustion. He begins by inscribing the largest triangle (triangle ABC) within the parabolic segment. The method proceeds by filling the remaining space with more triangles whereby each stage doubles the number of triangles. Archimedes determined that the total area of the triangles constructed at each stage is 1/4 of the area of the triangles constructed in the previous stage.

Let be the area of the first triangle (pictured in yellow), and be the area of the parabolic segment. Determine the ratio of the two areas .

【Solution】

Let represent the total area of the blue triangles, be the total area of the light red triangles , be the total area of the dark red triangles. Each stage is 1/4 the area of the previous. So,

This process continues indefinitely, generating the infinite geometric series:

Here, the first term is and the common ratio is . Using the infinite geometric formula gives:

Subsequently,

Therefore, the ratio of the two areas is

【Historical Note】

Archimedes was not aware of the infinite geometric series formula. He used proof by contradiction to arrive at the result！