To describe the sequence \(15, -45, 135, \ldots\), we can first look for a pattern in the terms.
- Finding the Ratio between Terms:
- The second term divided by the first term gives us: \[ \frac{-45}{15} = -3 \]
- The third term divided by the second term gives us: \[ \frac{135}{-45} = -3 \]
This tells us that each term is obtained by multiplying the previous term by \(-3\).
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General Form of the Sequence: This suggests that the sequence is geometric, where each term is derived from the first term through a common ratio of \(-3\).
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Finding the General Formula: The formula for the \(n\)-th term 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.
Here, \(a_1 = 15\) and \(r = -3\).
Thus, the \(n\)-th term can be expressed as: \[ a_n = 15 \cdot (-3)^{(n-1)} \]
So, the final equation to describe the sequence is: \[ a_n = 15 \cdot (-3)^{n-1} \]
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