The diagram shows a curved rod AB of length 100 cm which forms an arc of a circle. The end points A and B of the rod are 99 cm apart. The circle has radius r cm and the arc AB subtends an angle of 2α radians at O, the centre of the circle.

(i) Show that α satisfies the equation \(\frac{99}{100}x = \sin x\).

(ii) Given that this equation has exactly one root in the interval \(0 < x < \frac{1}{2} \pi\), verify by calculation that this root lies between 0.1 and 0.5.

(iii) Show that if the sequence of values given by the iterative formula \(x_{n+1} = 50 \sin x_n - 48.5 x_n\) converges, then it converges to a root of the equation in part (i).

(iv) Use this iterative formula, with initial value \(x_1 = 0.25\), to find α correct to 3 decimal places, showing the result of each iteration.