Question

Write out the first five terms of the sequence an=(−2)n
. Determine if the sequence is arithmetic or geometric, and then find the common ratio or difference. Which option below is accurate?

Option #1: The sequence is arithmetic, and the common difference is 6.

Option #2: The sequence is arithmetic, and the common difference is −2
.

Option #3: The sequence is geometric, and the common ratio is −2
.

Option #4: The sequence is geometric, and the common ratio is 2.

Answer 1

Let's begin by calculating the first five terms of the sequence defined by \( a_n = (-2)^n \):

1. For \( n = 1 \):
\[
a_1 = (-2)^1 = -2
\]
2. For \( n = 2 \):
\[
a_2 = (-2)^2 = 4
\]
3. For \( n = 3 \):
\[
a_3 = (-2)^3 = -8
\]
4. For \( n = 4 \):
\[
a_4 = (-2)^4 = 16
\]
5. For \( n = 5 \):
\[
a_5 = (-2)^5 = -32
\]

So, the first five terms of the sequence are:
\[
-2, 4, -8, 16, -32
\]

Now, let's determine if the sequence is arithmetic or geometric.

### Arithmetic Sequence
In an arithmetic sequence, the difference between consecutive terms is constant. Let's calculate the differences:

- \( 4 - (-2) = 4 + 2 = 6 \)
- \( -8 - 4 = -8 - 4 = -12 \)
- \( 16 - (-8) = 16 + 8 = 24 \)
- \( -32 - 16 = -32 - 16 = -48 \)

The differences are: \( 6, -12, 24, -48 \). Since the differences are not constant, the sequence is **not arithmetic**.

### Geometric Sequence
In a geometric sequence, the ratio between consecutive terms is constant. Let's calculate the ratios:

- \( \frac{4}{-2} = -2 \)
- \( \frac{-8}{4} = -2 \)
- \( \frac{16}{-8} = -2 \)
- \( \frac{-32}{16} = -2 \)

The ratios are all equal to \( -2 \). This indicates that the sequence is indeed a **geometric sequence** with a common ratio of \( -2 \).

### Conclusion
The correct option is:
**Option #3: The sequence is geometric, and the common ratio is \( -2 \).**