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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 2 / 18H / c / Solution

The balance recurrence is

π_{i} = π_{i + 1} + π_{0} q_{i + 1} .

Working down from

N - 1

gives

π_{i} = π_{0} P (T_{1} > i)

. Since the sum of these tails is

E [T_{1}]

, the unique invariant law is

π_{i} = \frac{P ( T _{1} > i )}{E [ T _{1} ]} = \frac{\sum _{j = i + 1}^{N} q _{j}}{\sum _{j = 1}^{N} j q _{j}} .

Solved by gpt-5.6-sol high.

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