Question

Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term.
-13, 26, -52, ...
Write your answer using decimals and integers.
an = ____ (___)^n-1

Answer 1

To find an equation that describes the given sequence \(-13, 26, -52, ...\), we first need to analyze the terms to identify a pattern.

1. **Identify the terms**:
- The first term (\(a_1\)) is -13.
- The second term (\(a_2\)) is 26.
- The third term (\(a_3\)) is -52.

2. **Determine if the sequence is geometric**:
- To check if this is a geometric sequence, we can look at the ratios:
\[
\frac{a_2}{a_1} = \frac{26}{-13} = -2
\]
\[
\frac{a_3}{a_2} = \frac{-52}{26} = -2
\]
- The ratio between consecutive terms is consistently \(-2\).

3. **General formula for a geometric sequence**:
- A geometric sequence can be represented by the formula:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where \(a_1\) is the first term and \(r\) is the common ratio.

4. **Substituting the values**:
- The first term \(a_1\) is \(-13\), and the common ratio \(r\) is \(-2\).
- Therefore, the equation for the \(n\)-th term of the sequence is:

\[
a_n = -13 \cdot (-2)^{n-1}
\]

So the final answer is:

\[
a_n = -13 \cdot (-2)^{n-1}
\]