In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio \( r \). Given the terms \( 12 \), \( b \), and \( 75 \), we can establish the following relationships:
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The relationship between the first and second terms can be expressed as: \[ b = 12r \]
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The relationship between the second and third terms can be expressed as: \[ 75 = br \]
Substituting the expression for \( b \) from the first equation into the second equation gives us: \[ 75 = (12r)r \] or \[ 75 = 12r^2. \]
To find \( r^2 \), we solve for \( r^2 \): \[ r^2 = \frac{75}{12} = \frac{25}{4}. \]
Taking the square root of both sides: \[ r = \sqrt{\frac{25}{4}} = \frac{5}{2}. \]
Thus, the common ratio \( r \) for the geometric sequence is: \[ \boxed{\frac{5}{2}}. \]