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.
-69, 138, -276,
Write your answer using decimals and integers.
an = __ ( __ )^n-1

Answer 1

To derive a formula for the sequence -69, 138, -276, we first need to identify a pattern in the sequence.

1. **Identify the first term**:
- The first term (when n = 1) is \( a_1 = -69 \).

2. **Identify the pattern** between consecutive terms:
- The second term (n = 2) is \( a_2 = 138 \).
- The third term (n = 3) is \( a_3 = -276 \).
- We can see that the sequence alternates in sign and appears to increase in absolute value.

3. **Calculate the ratios** of consecutive terms:
- \( \frac{a_2}{a_1} = \frac{138}{-69} = -2 \)
- \( \frac{a_3}{a_2} = \frac{-276}{138} = -2 \)

This indicates that each term is -2 times the previous term.

4. **Identify the general form of the sequence**:
- The sequence can be defined using an exponential function based on the previous calculations.

The general term can be written as:
\[ a_n = a_1 \cdot r^{(n-1)} \]
where \( r \) is the common ratio.

Substituting the known values:
- \( a_1 = -69 \)
- \( r = -2 \)

Therefore, the equation that describes the sequence is:
\[
a_n = -69 \cdot (-2)^{(n-1)}
\]

So, you can fill in the answer as:
\[
a_n = -69 \cdot (-2)^{n-1}
\]