To find the equation that describes the sequence \(3, -6, 12, \ldots\), we first observe the terms in the sequence:
- The first term (\(a_1\)) is \(3\).
- The second term (\(a_2\)) is \(-6\).
- The third term (\(a_3\)) is \(12\).
Next, we analyze the pattern in the sequence:
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Identify the pattern:
- From \(3\) to \(-6\), the value decreases by \(9\).
- From \(-6\) to \(12\), the value increases by \(18\).
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Finding a pattern in the differences:
- The first difference is:
- \(-6 - 3 = -9\)
- \(12 - (-6) = 18\)
- The differences appear to be alternating in sign and growing in magnitude.
- The first difference is:
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Finding the relationship:
- The terms of the sequence can also be calculated with a power of \(-2\):
- When \(n = 1\): \(3 = 3 \cdot (-2)^0\)
- When \(n = 2\): \(-6 = 3 \cdot (-2)^1\)
- When \(n = 3\): \(12 = 3 \cdot (-2)^2\)
- The terms of the sequence can also be calculated with a power of \(-2\):
Thus, we can observe that the formula for the \(n\)-th term is:
\[ a_n = 3 \cdot (-2)^{n-1} \]
Therefore, the final form of the equation that describes the sequence is:
\[ a_n = 3 \cdot (-2)^{n-1} \]