Question

Write out the first five terms of the sequence an=(−2)n
𝑎
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
−
2
.

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

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

(1 point)
Option # is accurate.

Answer 1

To find the first five terms of the sequence given by \( a_n = (-2)^n \):

1. **Calculate each term**:
- For \( n = 1 \): \( a_1 = (-2)^1 = -2 \)
- For \( n = 2 \): \( a_2 = (-2)^2 = 4 \)
- For \( n = 3 \): \( a_3 = (-2)^3 = -8 \)
- For \( n = 4 \): \( a_4 = (-2)^4 = 16 \)
- For \( n = 5 \): \( a_5 = (-2)^5 = -32 \)

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

2. **Determine if the sequence is arithmetic or geometric**:
- A sequence is **arithmetic** if the difference between consecutive terms is constant.
- \( 4 - (-2) = 4 + 2 = 6 \)
- \( -8 - 4 = -12 \)
- \( 16 - (-8) = 16 + 8 = 24 \)
- \( -32 - 16 = -48 \)

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

- A sequence is **geometric** if the ratio of consecutive terms is constant.
- \( \frac{4}{-2} = -2 \)
- \( \frac{-8}{4} = -2 \)
- \( \frac{16}{-8} = -2 \)
- \( \frac{-32}{16} = -2 \)

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

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