Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.
3 , –6 , 12 , ...
Write your answer using decimals and integers.

an = _(_)^ n^ -^ 1

Answer 1

To find the equation that describes the sequence \(3, -6, 12, \ldots\), we first observe the terms in the sequence:

- The first term (\(a_1\)) is \(3\).
- The second term (\(a_2\)) is \(-6\).
- The third term (\(a_3\)) is \(12\).

Next, we analyze the pattern in the sequence:

1. **Identify the pattern:**
- From \(3\) to \(-6\), the value decreases by \(9\).
- From \(-6\) to \(12\), the value increases by \(18\).

2. **Finding a pattern in the differences:**
- The first difference is:
- \(-6 - 3 = -9\)
- \(12 - (-6) = 18\)
- The differences appear to be alternating in sign and growing in magnitude.

3. **Finding the relationship:**
- The terms of the sequence can also be calculated with a power of \(-2\):
- When \(n = 1\): \(3 = 3 \cdot (-2)^0\)
- When \(n = 2\): \(-6 = 3 \cdot (-2)^1\)
- When \(n = 3\): \(12 = 3 \cdot (-2)^2\)

Thus, we can observe that the formula for the \(n\)-th term is:

\[
a_n = 3 \cdot (-2)^{n-1}
\]

Therefore, the final form of the equation that describes the sequence is:

\[
a_n = 3 \cdot (-2)^{n-1}
\]