Continued Fraction Fundamental Recurrence Relation

For a Simple Continued Fraction $\sigma=[a_0, a_1, \ldots]$ with Convergents $p_n/q_n$, the fundamental Recurrence Relation is given by

$$p_nq_{n-1}-p_{n-1}q_n=(-1)^n.$$

References

Olds, C. D. Continued Fractions. New York: Random House, p. 27, 1963.