In a geometric sequence, the common ratio \( r \) can be found by examining the relationship between consecutive terms. The general formula for the \( n \)-th term of a geometric sequence can be expressed as:
\[ a_n = r \cdot a_{n-1} \]
Given the recursively defined sequence, we have:
- \( a_1 = \frac{5}{6} \)
- \( a_n = 4a_{n-1} \)
To find the common ratio \( r \), we can write it in terms of the recursive definition:
From the definition,
\[ a_n = 4a_{n-1} \]
To relate this to the formula for a geometric sequence, we can express the common ratio as \( r = \frac{a_n}{a_{n-1}} \).
Substituting with the given recursive formula:
\[ r = \frac{a_n}{a_{n-1}} = \frac{4a_{n-1}}{a_{n-1}} = 4 \]
Thus, the common ratio \( r \) is:
\[ \boxed{4} \]