In a geometric sequence, the general formula for the \(n\)-th term \(a_n\) is given by:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.
Given that the first term \(a_1 = -3\) and the common ratio \(r = \frac{1}{2}\), we can write the equation for the \(n\)-th term as follows:
- The equation for the geometric sequence is:
\[ a_n = -3 \cdot \left(\frac{1}{2}\right)^{(n-1)} \]
Now, to find the 6th term of the sequence, we substitute \(n = 6\) into the equation:
- Calculate \(a_6\):
\[ a_6 = -3 \cdot \left(\frac{1}{2}\right)^{(6-1)} = -3 \cdot \left(\frac{1}{2}\right)^{5} \]
Now, calculate \(\left(\frac{1}{2}\right)^{5}\):
\[ \left(\frac{1}{2}\right)^{5} = \frac{1}{2^5} = \frac{1}{32} \]
Now substitute back into the equation:
\[ a_6 = -3 \cdot \frac{1}{32} = -\frac{3}{32} \]
Therefore, the 6th term of the sequence is:
\[ \boxed{-\frac{3}{32}} \]
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