To find the recursive formula of the given geometric sequence \(0.2, -1, 5, -25, \ldots\), let's first identify the first term and the common ratio.
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The first term is: \[ a(1) = 0.2 \]
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To find the common ratio, we can divide any term by the previous term:
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To find the ratio between the second term and the first term: \[ r = \frac{-1}{0.2} = -5 \]
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To verify, we can check the ratio between the third term and the second term: \[ r = \frac{5}{-1} = -5 \]
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Finally, let's check the ratio between the fourth term and the third term: \[ r = \frac{-25}{5} = -5 \]
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Since the ratio is consistent across the terms, we confirm the common ratio \(r = -5\).
Putting this information together, we can write the recursive formula for the sequence:
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The first term: \[ a(1) = 0.2 \]
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The recursive relationship: \[ a(n) = a(n-1) \cdot (-5) \]
Therefore, the completed recursive formula is:
\[ \begin{align*} a(1) &= 0.2, \ a(n) &= a(n-1) \cdot (-5) \text{ for } n > 1. \end{align*} \]
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