To find the equation for the given sequence \(-28, -84, -252\), we first need to determine whether it's an arithmetic or geometric sequence.
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Identify the Sequence Type:
- Common Ratio: Let's see if this sequence is geometric by calculating the ratio between terms. \[ \frac{-84}{-28} = 3 \] \[ \frac{-252}{-84} = 3 \] Since the ratio between consecutive terms is consistent (both are 3), this is a geometric sequence.
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General Form of a Geometric Sequence: The general form of a geometric sequence can be expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \(a_1\) is the first term and \(r\) is the common ratio.
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Identify the Constants:
- The first term \(a_1 = -28\)
- The common ratio \(r = 3\)
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Write the Equation: Now we can substitute these values into the equation: \[ a_n = -28 \cdot (3)^{(n-1)} \]
Thus, the equation for the sequence is: \[ \boxed{a_n = -28 \cdot (3)^{(n-1)}} \]