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.
-6, 12, -24,
Write your answer using decimals and integers.
an = __ ( __ )^n-1

Answer 1

To describe the sequence \(-6, 12, -24\), we can first observe the pattern in the terms:

1. The first term, \( a_1 = -6 \).
2. The second term, \( a_2 = 12 \).
3. The third term, \( a_3 = -24 \).

Next, we can analyze the relationship between the terms:
- From \( -6 \) to \( 12 \): The second term is \( -6 \times -2 = 12 \).
- From \( 12 \) to \( -24 \): The third term is \( 12 \times -2 = -24 \).

We observe that the terms alternate in sign and each term is multiplied by \(-2\) to get the next term.

This indicates that the sequence is geometric with a common ratio \( r = -2 \).

The general formula for a geometric sequence is:
\[
a_n = a_1 \cdot r^{n-1}
\]
Where:
- \( a_1 \) is the first term
- \( r \) is the common ratio
- \( n \) is the position in the sequence

Now substituting in our values:
- \( a_1 = -6 \)
- \( r = -2 \)

The formula for this sequence can thus be expressed as:
\[
a_n = -6 \cdot (-2)^{n-1}
\]

So, in the desired format:
\[
a_n = -6 \cdot (-2)^{n-1}
\]