To find the common difference in the sequence \( \frac{3}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{3}{2} \), you need to subtract each term from the term that follows it.
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The difference between the second term and the first term: \[ \frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2} = \frac{-2}{2} = -1 \]
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The difference between the third term and the second term: \[ -\frac{1}{2} - \frac{1}{2} = -\frac{1 + 1}{2} = -\frac{2}{2} = -1 \]
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The difference between the fourth term and the third term: \[ -\frac{3}{2} - \left(-\frac{1}{2}\right) = -\frac{3}{2} + \frac{1}{2} = -\frac{3 - 1}{2} = -\frac{2}{2} = -1 \]
In all cases, the common difference is \(-1\).
Thus, the common difference of the sequence is \(-1\).