Question

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.


–2 , –8 , –32 , ...


Write your answer using decimals and integers.

an =
(
)n - 1

Answer 1

To identify the equation for the sequence \(-2, -8, -32, ...\), let's first analyze the pattern in the sequence.

1. **Identifying the pattern**:
- The first term (\(a_1\)) is \(-2\)
- The second term (\(a_2\)) is \(-8\)
- The third term (\(a_3\)) is \(-32\)

2. **Finding the common ratio**:
To find how each term relates to the previous term, we can look at the ratios:
- \(\frac{a_2}{a_1} = \frac{-8}{-2} = 4\)
- \(\frac{a_3}{a_2} = \frac{-32}{-8} = 4\)

The ratio is constant at 4, indicating that this is a geometric sequence.

3. **General formula**:
The general formula for a geometric sequence can be expressed as:
\[
a_n = a_1 \cdot r^{(n-1)}
\]
where:
- \(a_1\) is the first term,
- \(r\) is the common ratio,
- \(n\) is the term number.

In our case:
- \(a_1 = -2\)
- \(r = 4\)

Putting it all together, we have:
\[
a_n = -2 \cdot 4^{(n-1)}
\]

So the final equation describing the sequence is:
\[
a_n = -2 \cdot 4^{(n-1)}
\]