Let's first calculate the first five terms of the sequence \( a_n = (-2)^n \):
- 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 \( n = 5 \): \( a_5 = (-2)^5 = -32 \)
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.
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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.
-
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.