Question

What is the recursive formula for the geometric sequence with this explicit formula?

a(n) = 6 • (- 1/4) ^(n-1)

Answer 1

To find the recursive formula for the geometric sequence given by the explicit formula \( a(n) = 6 \cdot \left(-\frac{1}{4}\right)^{n-1} \), we start by identifying the first term and the common ratio.

1. **Identify the first term (\(a(1)\))**:
\[
a(1) = 6 \cdot \left(-\frac{1}{4}\right)^{1-1} = 6 \cdot 1 = 6
\]

2. **Determine the common ratio (\(r\))**:
The common ratio \(r\) is the factor by which we multiply to get from one term to the next. In this case, we can see from the explicit formula that:
\[
r = -\frac{1}{4}
\]

3. **Write the recursive formula**:
A recursive formula for a sequence typically has the form:
\[
a(n) = a(n - 1) \cdot r
\]
for \(n > 1\), along with the first term.

Thus, using the values we found:

- The first term \(a(1) = 6\)
- The common ratio \(r = -\frac{1}{4}\)

The recursive formula is:
\[
\begin{cases}
a(1) = 6 \\
a(n) = a(n - 1) \cdot \left(-\frac{1}{4}\right) \text{ for } n > 1
\end{cases}
\]