Complete the recursive formula of the geometric sequence

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.

1. The first term is:
\[
a(1) = 0.2
\]

2. To find the common ratio, we can divide any term by the previous term:

- To find the ratio between the second term and the first term:
\[
r = \frac{-1}{0.2} = -5
\]

- To verify, we can check the ratio between the third term and the second term:
\[
r = \frac{5}{-1} = -5
\]

- Finally, let's check the ratio between the fourth term and the third term:
\[
r = \frac{-25}{5} = -5
\]

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:

- The first term:
\[
a(1) = 0.2
\]

- The recursive relationship:
\[
a(n) = a(n-1) \cdot (-5)
\]

Therefore, the completed recursive formula is:

\[
\begin{align*}
a(1) &= 0.2, \\
a(n) &= a(n-1) \cdot (-5) \text{ for } n > 1.
\end{align*}
\]