To find the equation that describes the sequence \( 8, -40, 200, \ldots \), we need to identify a pattern in the terms.
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Finding the pattern: Let's look at the given terms individually:
- First term (\(a_1\)): 8
- Second term (\(a_2\)): -40
- Third term (\(a_3\)): 200
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Finding the ratios:
- From \(8\) to \(-40\): \(-40 / 8 = -5\)
- From \(-40\) to \(200\): \(200 / -40 = -5\)
The ratio of each term to its previous term seems to be \(-5\), so it suggests that each term is being multiplied by \(-5\).
- Expressing the general term:
- The first term is \(a_1 = 8\).
- Each subsequent term can be calculated as: \[ a_n = a_{n-1} \cdot (-5) \]
- Therefore, you can express this using the first term: \[ a_n = 8 \cdot (-5)^{n-1} \]
Putting it all together, the equation that describes the nth term of the sequence is:
\[ a_n = 8 \cdot (-5)^{n-1} \]
This represents the sequence you provided.