2017 RI Promo Q3

A piece of paper in the form of a semi-circle of radius \(r\) is cut into twelve sectors such that the areas are in arithmetic progression, and the area of the biggest sector is three times that of the smallest sector.

Find the exact area of the smallest sector in terms of \(r\).[2]

An arithmetic series \(A\) has first term a and common difference \(d\), where \(a\) and \(d\) are non-zero. A convergent geometric series \(G\) has common ratio \(r\).

The first three terms of \(G\) are equal to the first, eleventh and seventeenth terms of \(A\), respectively.