Let's start with the given sequence \( a_n = (-2)^n \). We'll calculate the first five terms:
- 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 \( n = 5 \): \( a_5 = (-2)^5 = -32 \)
Thus, the first five terms of the sequence are:
\(-2, 4, -8, 16, -32\)
Next, we will determine if this sequence is arithmetic or geometric.
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Arithmetic Sequence: This type of sequence has a constant difference between consecutive terms.
- 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, this sequence is not arithmetic.
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Geometric Sequence: This type of sequence has a constant ratio between consecutive terms.
- Ratio between \( a_2 \) and \( a_1 \): \( \frac{4}{-2} = -2 \)
- Ratio between \( a_3 \) and \( a_2 \): \( \frac{-8}{4} = -2 \)
- Ratio between \( a_4 \) and \( a_3 \): \( \frac{16}{-8} = -2 \)
- Ratio between \( a_5 \) and \( a_4 \): \( \frac{-32}{16} = -2 \)
Since the ratio is constant (\(-2\)), this sequence is geometric, and the common ratio is \(-2\).
Based on this analysis, the correct option is:
Option #3: The sequence is geometric, and the common ratio is −2.