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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 3 / 11G / a / Solution

With

p_{- 1} = 1, p_{0} = a_{0}

,

q_{- 1} = 0, q_{0} = 1

and

p_{n} = a_{n} p_{n - 1} + p_{n - 2}

,

q_{n} = a_{n} q_{n - 1} + q_{n - 2}

, induction gives

p_{n} q_{n - 1} - p_{n - 1} q_{n} = (- 1)^{n + 1}

. Matrix multiplication gives

[a_{0}, \dots, a_{n}, β] = (p_{n} β + p_{n - 1}) / (q_{n} β + q_{n - 1})

; monotonicity in

β > 0

places it strictly between the adjacent convergents.

Solved by gpt-5.6-sol high.

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