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.

(1 point)
Option #
is accurate.

Answer 1

Let's first calculate the first five terms of the sequence \( a_n = (-2)^n \):

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 \( n = 5 \): \( a_5 = (-2)^5 = -32 \)

The first five terms of the sequence are:
- \( a_1 = -2 \)
- \( a_2 = 4 \)
- \( a_3 = -8 \)
- \( a_4 = 16 \)
- \( a_5 = -32 \)

Thus, the sequence is: \(-2, 4, -8, 16, -32\).

Next, we determine if the sequence is arithmetic or geometric.

1. **Arithmetic Sequence**: The difference between consecutive terms must be constant.
- \( 4 - (-2) = 6 \)
- \( -8 - 4 = -12 \)
- \( 16 - (-8) = 24 \)
- \( -32 - 16 = -48 \)

The differences between the terms are not constant, so the sequence is **not arithmetic**.

2. **Geometric Sequence**: The ratio of consecutive terms must be constant.
- \( \frac{4}{-2} = -2 \)
- \( \frac{-8}{4} = -2 \)
- \( \frac{16}{-8} = -2 \)
- \( \frac{-32}{16} = -2 \)

The ratios between the terms are all -2, so the sequence is **geometric**.

The common ratio \( r \) is \(-2\).

Thus, the correct option is:

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