1. The problem asks for the common ratio $r$ of the geometric sequence $(42, 21, 10.5, 5.25, \ldots)$.\n\n2. The common ratio $r$ in a geometric sequence is found by dividing any term by the previous term: $$r = \frac{a_{n+1}}{a_n}$$\n\n3. Calculate $r$ using the first two terms: $$r = \frac{21}{42}$$\n\n4. Simplify the fraction by dividing numerator and denominator by their greatest common divisor 21: $$r = \frac{\cancel{21}}{\cancel{42}} = \frac{1}{2}$$\n\n5. To confirm, check the ratio between the next terms: $$\frac{10.5}{21} = \frac{1}{2}$$ and $$\frac{5.25}{10.5} = \frac{1}{2}$$, confirming the ratio is consistent.\n\n6. Therefore, the common ratio of the sequence is $$r = \frac{1}{2}$$.