Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 11F / iv / Solution

The denominators of the convergents areThey 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 $p_x/q_x=34/13$.