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 find the first five terms of the sequence given 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:

\[
-2, 4, -8, 16, -32
\]

Next, we need to determine whether the sequence is arithmetic or geometric.

**Arithmetic Sequence:** A sequence is arithmetic if the difference between consecutive terms is constant.
- Difference between \( a_2 \) and \( a_1 \): \( 4 - (-2) = 4 + 2 = 6 \)
- Difference between \( a_3 \) and \( a_2 \): \( -8 - 4 = -12 \)
- Difference between \( a_4 \) and \( a_3 \): \( 16 - (-8) = 16 + 8 = 24 \)
- Difference between \( a_5 \) and \( a_4 \): \( -32 - 16 = -48 \)

Since the differences are not constant, the sequence is **not arithmetic**.

**Geometric Sequence:** A sequence is geometric if the ratio of consecutive terms is constant.
- Ratio of \( a_2 \) to \( a_1 \): \( \frac{4}{-2} = -2 \)
- Ratio of \( a_3 \) to \( a_2 \): \( \frac{-8}{4} = -2 \)
- Ratio of \( a_4 \) to \( a_3 \): \( \frac{16}{-8} = -2 \)
- Ratio of \( a_5 \) to \( a_4 \): \( \frac{-32}{16} = -2 \)

Since the ratios are constant (\(-2\)), the sequence is **geometric** with a common ratio of \(-2\).

Based on the options provided:

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