 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-2/10e/b/i/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

h (x)

x \in {0, \dots, n}

hits zero before

n

. The first-step recurrence is

h (x) = \frac{1}{2} h (x - 1) + \frac{1}{2} h (x + 1), h (0) = 1, h (n) = 0.

The recurrence says that

h

is affine, and the boundary values determine it:

h (x) = \frac{n - x}{n} .

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