To find the common ratio of the given recursively defined geometric sequence, we start with the first term provided, \( a_1 = \frac{5}{6} \), and the recursive definition \( a_n = 4a_{n-1} \).
In a geometric sequence, the common ratio \( r \) can be determined by the relationship of consecutive terms, i.e., \( r = \frac{a_n}{a_{n-1}} \).
Using the recursive definition: \[ a_2 = 4a_1 \] We calculate \( a_2 \): \[ a_2 = 4 \cdot \frac{5}{6} = \frac{20}{6} = \frac{10}{3} \]
Now we can calculate the common ratio \( r \): \[ r = \frac{a_2}{a_1} = \frac{\frac{10}{3}}{\frac{5}{6}} = \frac{10}{3} \cdot \frac{6}{5} = \frac{10 \cdot 6}{3 \cdot 5} = \frac{60}{15} = 4 \]
The calculation shows that: \[ r = 4 \]
Thus, the common ratio of the given geometric sequence is \(\boxed{4}\).