Sequence and Series - Proving the progression

The diagram shows a square \(\mathrm{ABCD}\) of side \(4\) cm. The midpoints \(\mathrm{P, Q, R, S}\) of the sides are joined to form a second square.

1. 1. Show that \(\mathrm{PQ} = 2\sqrt{2}\) cm.
   2. Find the area of \(\mathrm{PQRS}\).[3]

The midpoints \(\mathrm{W, X, Y, Z}\) of the sides of \(\mathrm{P, Q, R, S}\) are now joined to form a third square as shown.

2. 1. Write down the area of the third square, \(\mathrm{WXYZ}\).
   2. Show that the areas of \(\mathrm{ABCD}\), \(\mathrm{PQRS}\) and \(\mathrm{WXYZ}\) form a geometric sequence. Find the common ratio of this sequence.[3]

The process of forming smaller and squares (by joining the midpoints) is continued indefinitely.

3. 1. Find the area of the \(11^{\mathrm{th}}\) square.
   2. Calculate the sum of the areas of all the squares.