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Question id: 50. This question is similar to one that appeared in an IB Standard paper in 2014. The use of a calculator is not allowed.

Sequences

A square is drawn with sides of length 32 cm. The midpoints of the sides of this square are joined to form a new square and four red triangles. The process is repeated to produce yellow triangles and then again to produce blue triangles.

The length of the equal sides of the red triangles are denoted by \(x_1\) and their areas are each \(A_1\).
The length of the equal sides of the yellow triangles are denoted by \(x_2\) and their areas are each \(A_2\).
The length of the equal sides of the blue triangles are denoted by \(x_3\) and their areas are each \(A_3\).

(a) The following table gives the values of \(x_n\) and \(A_n\), for \(1\le n\le3\). Copy and complete the table.

\(n\) 1  2  3 
\(x_n\)16      
\(A_n\)128  

(b) The process of drawing smaller and smaller squares inside each new square is repeated. Find \(A_7\)

(c) Consider an initial square of side length \(k\) cm. The process described above is repeated indefinitely. The total area of one of each colour triangles is \(k\) cm2. Find the value of \(k\).

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