To find the common ratio of the geometric sequence defined by \( a_1 = \frac{5}{6} \) and \( a_n = 4a_{n-1} \), we start with the recursive formula:
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Identify the first term: \[ a_1 = \frac{5}{6} \]
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Use the recursive formula to find the next term, \( a_2 \): \[ a_2 = 4a_1 = 4 \left(\frac{5}{6}\right) = \frac{20}{6} = \frac{10}{3} \]
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Now, consider the ratio between consecutive terms: The common ratio \( r \) in a geometric sequence is defined as: \[ r = \frac{a_n}{a_{n-1}} \] Specifically for this sequence: \[ r = \frac{a_2}{a_1} = \frac{\frac{10}{3}}{\frac{5}{6}} = \frac{10}{3} \times \frac{6}{5} = \frac{60}{15} = 4 \]
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Conclusion: The common ratio \( r \) for the given geometric sequence is \( 4 \).
Therefore, the answer is: \[ \boxed{4} \]