To identify the equation for the sequence \(-2, -8, -32, ...\), let's first analyze the pattern in the sequence.
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Identifying the pattern:
- The first term (\(a_1\)) is \(-2\)
- The second term (\(a_2\)) is \(-8\)
- The third term (\(a_3\)) is \(-32\)
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Finding the common ratio: To find how each term relates to the previous term, we can look at the ratios:
- \(\frac{a_2}{a_1} = \frac{-8}{-2} = 4\)
- \(\frac{a_3}{a_2} = \frac{-32}{-8} = 4\)
The ratio is constant at 4, indicating that this is a geometric sequence.
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General formula: The general formula for a geometric sequence can be expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.
In our case:
- \(a_1 = -2\)
- \(r = 4\)
Putting it all together, we have: \[ a_n = -2 \cdot 4^{(n-1)} \]
So the final equation describing the sequence is: \[ a_n = -2 \cdot 4^{(n-1)} \]