Infinite sum of segments equals 2: \( p + p^2 + p^3 + \cdots = 2 \) (A1)

Recognize geometric series with first term \( p \), common ratio \( p \) (M1)

Use sum formula: \( S_\infty = \frac{p}{1 – p} \) (A1)

Set up equation: \( \frac{p}{1 – p} = 2 \) (A1)

Solve: \( p = 2 – 2p \implies 3p = 2 \implies p = \frac{2}{3} \) (A1) (AG)

[5 marks]

Recognize infinite geometric series for square areas: \( k^2 + k^4 + k^6 + \cdots \) (M1)

Use sum formula: \( S_\infty = \frac{k^2}{1 – k^2} = \frac{9}{16} \) (A2)

Solve: \( 16k^2 = 9 – 9k^2 \implies 25k^2 = 9 \implies k^2 = \frac{9}{25} \implies k = \frac{3}{5} \) (A1, A1)

Recognize segment lengths along [CD]: \( k + k^2 + k^3 + \cdots = b \) (M1)

Use sum formula: \( b = \frac{k}{1 – k} \) (A1)

Substitute \( k = \frac{3}{5} \): \( b = \frac{\frac{3}{5}}{1 – \frac{3}{5}} = \frac{\frac{3}{5}}{\frac{2}{5}} = \frac{3}{2} \) (M1, A1) (N3)

Answer: \( b = \frac{3}{2} \) [9 marks]