Question

Find an explicit formula for the geometric sequence

\[\dfrac{1}2\,,-4\,,\,32\,,-256,..\].
Note: the first term should be
\[\text{\it{a(1)}}\].
\[a(n)=\]

Answer 1

To find the explicit formula for the given geometric sequence \(\frac{1}{2}, -4, 32, -256, \ldots\), we need to first identify the first term and the common ratio of the sequence.

1. **Identifying the first term**:
The first term is:
\[
a(1) = \frac{1}{2}
\]

2. **Finding the common ratio**:
To find the common ratio \(r\), we divide the second term by the first term:
\[
r = \frac{-4}{\frac{1}{2}} = -4 \times 2 = -8
\]

We can verify this ratio by checking the subsequent terms:
- From the second term to the third term:
\[
r = \frac{32}{-4} = -8
\]
- From the third term to the fourth term:
\[
r = \frac{-256}{32} = -8
\]

Thus, the common ratio \(r\) is indeed \(-8\).

3. **General formula for a geometric sequence**:
The explicit formula for the \(n\)-th term of a geometric sequence is given by:
\[
a(n) = a(1) \cdot r^{n-1}
\]

Substituting the values we found:
- \(a(1) = \frac{1}{2}\)
- \(r = -8\)

We get:
\[
a(n) = \frac{1}{2} \cdot (-8)^{n-1}
\]

Therefore, the explicit formula for the geometric sequence is:
\[
\boxed{a(n) = \frac{1}{2} \cdot (-8)^{n-1}}
\]