Let's first find the first five terms of the sequence given by \( a_n = (-2)^n \):
- 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:
\[ -2, 4, -8, 16, -32 \]
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.