 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-4/11f/iv/solution

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

The denominators of the convergents are

1, 1, 2, 3, 14, 17, 31, 48, \dots .

They increase strictly from

q_{2}

onward, and none equals

13

. Equivalently,

34/13 = [2; 1, 1, 1, 1, 2]

, which ceases to agree with

[2; \overline{1, 1, 1, 4}]

after the third partial quotient. Hence there is no positive index

x

with

p_{x} / q_{x} = 34/13

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