Difference between Adjacent Convergents But One of Simple Continued Fraction

Theorem
Then:
 * $p_k q_{k - 2} - p_{k - 2} q_k = \left({-1}\right)^{k - 1} a_k$

That is:
 * $C_k - C_{k-2} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 2} } {q_{k - 2} } = \dfrac {\left({-1}\right)^{k - 1} a_k} {q_k q_{k - 2} }$

for $k \ge 3$.