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Pure periodicity means that $\theta$ reappears as the tail after one period. The continued-fraction tail formula therefore givesAfter cross-multiplication,

Now letBy part (c), the matrix for one reversed period isso its positive fixed point $\eta$ satisfiesPutting $\eta =-1/X$ and multiplying by $-X^2$ turns this equation into $f(X)=0$. The product of the two roots of $f$ is $-p_{n-1}/q_n<0$; because $\theta >0$, the other root ${\theta }^{\prime }$ is negative. Consequently $-1/{\theta }^{\prime }>0$ is the positive fixed point selected by convergence of the reversed continued fraction. The reversed-period identity for a purely periodic continued fraction is therefore