To determine the equation for the sequence \(-6, -18, -54, \ldots\), we first need to identify the pattern in the terms.
- Identify the relationship between terms:
- The first term (\(a_1\)) is \(-6\).
- The second term (\(a_2\)) is \(-18\) which is \(-6 \times 3\).
- The third term (\(a_3\)) is \(-54\) which is \(-18 \times 3\).
From this, we can conclude that each term is obtained by multiplying the previous term by \(3\).
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Recognizing the sequence type: This is a geometric sequence where each term after the first can be represented as: \[ a_n = a_1 \cdot r^{(n-1)} \] where:
- \(a_1 = -6\) (the first term),
- \(r = 3\) (the common ratio).
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Substituting into the formula: Substituting the values we found: \[ a_n = -6 \cdot 3^{(n-1)} \]
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Final equation: Thus, the equation to describe the sequence is given by: \[ a_n = -6 \cdot 3^{(n-1)} \]