Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


–17 , 34 , –68 , ...


Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

To find an equation that describes the given sequence \(-17, 34, -68, \ldots\), we can first observe the pattern in the sequence.

1. **Recognizing the Pattern**:
- The first term (\(n = 1\)) is \(-17\).
- The second term (\(n = 2\)) is \(34\).
- The third term (\(n = 3\)) is \(-68\).

2. **Finding the Pattern**:
- Notice that the absolute values of the terms seem to follow a multiplication pattern:
- \(17 \times 2^0 = 17\)
- \(17 \times 2^1 = 34\)
- \(17 \times 2^2 = 68\)
- Additionally, the signs alternate. The first term is negative, the second is positive, and the third is negative.

3. **General Formula**:
- The absolute value of the \(n\)-th term can be formulated as:
\[
|a_n| = 17 \cdot 2^{n-1}
\]
- The sign can be expressed as \((-1)^n\):
- For \(n = 1\), \((-1)^1 = -1\)
- For \(n = 2\), \((-1)^2 = 1\)
- For \(n = 3\), \((-1)^3 = -1\)

4. **Putting It All Together**:
- Combining both parts, we have:
\[
a_n = -17 \cdot 2^{n-1} \cdot (-1)^{n-1}
\]

This can also be simplified. Specifically, the effect of the \((-1)^{n-1}\) on the initial negative sign results in the correct alternating signs.

Hence, the equation to describe the sequence is:
\[
a_n = -17 \cdot (-2)^{n-1}
\]
This correctly reflects both the values and alternating signs of the sequence.

Thus the final answer is:
\[
a_n = -17 \cdot (-2)^{n-1}
\]