Let's begin by calculating the first five terms of the sequence defined 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: \[ -2, 4, -8, 16, -32 \]
Now, let's determine if the sequence is arithmetic or geometric.
Arithmetic Sequence
In an arithmetic sequence, the difference between consecutive terms is constant. Let's calculate the differences:
- \( 4 - (-2) = 4 + 2 = 6 \)
- \( -8 - 4 = -8 - 4 = -12 \)
- \( 16 - (-8) = 16 + 8 = 24 \)
- \( -32 - 16 = -32 - 16 = -48 \)
The differences are: \( 6, -12, 24, -48 \). Since the differences are not constant, the sequence is not arithmetic.
Geometric Sequence
In a geometric sequence, the ratio between consecutive terms is constant. Let's calculate the ratios:
- \( \frac{4}{-2} = -2 \)
- \( \frac{-8}{4} = -2 \)
- \( \frac{16}{-8} = -2 \)
- \( \frac{-32}{16} = -2 \)
The ratios are all equal to \( -2 \). This indicates that the sequence is indeed a geometric sequence with a common ratio of \( -2 \).
Conclusion
The correct option is: Option #3: The sequence is geometric, and the common ratio is \( -2 \).
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