Sequence and Series - Proving the progression

The diagrams below show the first four squares in a sequence of squares which are subdivided in half. The area of the shaded square \(\mathrm{A}\) is \(\frac{1}{4}\)

1. 1. Find the area of square \(\mathrm{B}\) and of square \(\mathrm{C}\).
   2. Show that the areas of squares \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are in geometric progression.
   3. Write down the common ratio of the progression.[5]
2. 1. Find the total area shaded in diagram \(2\).
   2. Find the total area shaded in the \(8^{\mathrm{th}}\) diagram of this sequence.
      Give your answer correct to six significant figures.[4]
3. The dividing and shading process illustrated is continued indefinitely.
   Find the total area shaded.[2]