To find the equation that describes the given sequence \(-8, -32, -128, \ldots\), we first need to identify the pattern in the terms.
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Let's observe the sequence:
- First term (\(a_1\)): \(-8\)
- Second term (\(a_2\)): \(-32\)
- Third term (\(a_3\)): \(-128\)
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Now let's try to find a common ratio by dividing consecutive terms:
- \(\frac{a_2}{a_1} = \frac{-32}{-8} = 4\)
- \(\frac{a_3}{a_2} = \frac{-128}{-32} = 4\)
This shows that the sequence is geometric with a common ratio of \(4\).
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The first term can also be expressed as \(-8\), and since this is a geometric sequence, we can represent any term in the sequence using the formula for a geometric series: \[ a_n = a_1 \cdot r^{(n-1)} \] where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term.
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Plugging in the values:
- \(a_1 = -8\)
- \(r = 4\)
Thus, the formula becomes: \[ a_n = -8 \cdot 4^{(n-1)} \]
Therefore, the final equation that describes the sequence is: \[ a_n = -8(4)^{n-1} \]
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