Let $(p,q)$ be the fundamental solution to Pell's equation $x^2-dy^2=1$, which makes $\frac{p}{q}$ a convergent of $\sqrt{d}$, per the theorem underlying the continued fraction algorithm for solving Pell's equation.
Is it always (or with finitely many exceptions) the case that at least one of the reduced fractions $\frac{p+1}{q},\frac{p-1}{q}$ is also a convergent of $\sqrt{d}$?
Examples: $$ d=73\\ 2281249^2-267000^2d=1\\ \frac{2281249-1}{267000}=\frac{1068}{125},\ a\ convergent\ of\ \sqrt{d} $$
$$ d=95\\ 39^2-4^2d=1\\ \frac{39+1}{4}=\frac{10}{1},\ a\ convergent\ of\ \sqrt{d} $$
$$ d=96\\ 49^2-5^2d=1\\ \frac{49+1}{5}=\frac{10}{1},\ a\ convergent\ of\ \sqrt{d} $$